Embeddings of plane continua and the fixed point property

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Embeddings of plane continua and the fixed point property
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vi, 247 leaves : ill. ; 28 cm.
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Mayer, John Clyde, 1945-
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Topological imbeddings   ( lcsh )
Fixed point theory   ( lcsh )
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Mathematics thesis Ph. D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Includes bibliographical references (leaves 244-246).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by John Clyde Mayer.

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
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    Abstract
        Page v
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    Chapter 1. Introduction: The fixed point problem for nonseparating plane continua
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    Chapter 2. Embeddings and prime end structure of chainable continua
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    Chapter 3. Principal embeddings of atriodic plane continua
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    Chapter 4. Inequivalent embeddings and prime ends
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    Chapter 5. The prime end structure of quotient spaces
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    Chapter 6. Principal embeddings of plane continua and extendable homeomorphisms
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    References
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    Biographical sketch
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Full Text












EMBEDDINGS OF PLANE CONTINUE
AND THE FIXED POINT PROPERTY






BY

JOHN CLYDE MAYER



























DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1982














ACKNOWLEDGEMENT

Grateful acknowledgment is made for the untiring

assistance and direction of the author's advisor, Beverly L.

Brechner, in bringing this study to completion.
















TABLE OF CONTENTS


ACKNOWLEDGEMENT ......................................... ii

ABSTRACT ............................................... v


1 INTRODUCTION: THE FIXED POINT PROBLEM FOR
NONSEPARATING PLANE CONTINUE .................. 1

1.1 Motivation ................................ 1
1.2 Embeddings and Prime End Structure..... 3
1.3 Lake-of-Wada Channels.................... 4

2 EMBEDDINGS AND PRIME END STRUCTURE OF
CHAINABLE CONTINUA............................. 7

2.1 Introduction............................ 7
2.2 Embeddings of the Knaster U-Continuum.. 9
2.3 Principal Embeddings of Chainable
Continua.............................. 19
2.4 Nonprincipal Embeddings of Chainable
Continua.............................. . 23
2.5 N-Principal Embeddings of Chainable
Continua.............................. . 31

3 PRINCIPAL EMBEDDINGS OF ATRIODIC PLANE
CONTINUA ...................................... 53

3.1 Introduction........... ....... ........ 53
3.2 The X-odic Continuum.................... 54
3.3 Lake-of-Wada Channels................... 84

4 INEQUIVALENT EMBEDDINGS AND PRIME ENDS........ 87

4.1 Introduction............................ 87
4.2 The Sin 1/x Continuum................... 88
4.3 The Knaster U-Continuum................. 99
4.4 Uncountably Many Embeddings of
Uncountably Many Continua............. 127








5 THE PRIME END STRUCTURE OF QUOTIENT SPACES.... 145

5.1 Introduction........................... 145
5.2 Prime Ends and Quotient Spaces......... 148
5.3 Accessibility ........................... 172
5.4 The Psuedo Arc......................... 178
5.5 Chainable Continua with End Subcontinua 185
5.6 Inaccessibility......................... 195

6 PRINCIPAL EMBEDDINGS OF PLANE CONTINUE AND
EXTENDABLE HOMEOMORPHISMS...................... 223

6.1 Introduction............................ 223
6.2 Principal Embeddings of Triodic
Continue .............................. 224
6.3 Extendable Homeomorphisms of
Principally Embedded Chainable
Continua.............................. 233

REFERENCES.............................................. 245

BIOGRAPHICAL SKETCH..................................... 247












Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EMBEDDINGS OF PLANE CONTINUE
AND THE FIXED POINT PROPERTY

By

John Clyde Mayer

May 1982



Chairman: Beverly L. Brechner
Major Department: Mathematics

A long-outstanding problem in plane topology is the

following: Do nonseparating plane continue have the fixed

point property for continuous maps (Problem 107 in the

Scottish Problem Book)? A number of partial solutions are

known. One important result is a theorem due independently

to H. Bell and K. Sieklucki which has the following

consequence (pointed out by B. Brechner): A nonseparating

plane continuum admitting a fixed point free map must have a

Lake-of-Wada channel in its boundary in every embedding into

the plane. This consequence, which seems to reveal much

about the structure of any potential counter-example to the

fixed point property, motivates the studies of embeddings of

plane continue in this essay. The principal tool we utilize

to study such embeddings is prime end theory.

v










Major divisions of this dissertation are devoted to

(1) Embeddings of chainable continue with (one or more)

Lake-of-Wada channels, and theorems concerning the

conditions under which a chainable continuum can be

(re-)embedded without a Lake-of-Wada channel.

(2) An example of an atriodic nonchainable plane

continuum, embedded with a Lake-of-Wada channel, of which

type any minimal tree-like counter-example to the fixed

point property (should one exist) is likely to be.

(3) Embeddings of triodic tree-like continue with

(multiple) Lake-of-Wada channels.

(4) Inequivalent embeddings of plane continue and the

prime end structures of such embeddings, including the fact

that the sin 1/x continuum and the Knaster U-continuum each

have uncountably many inequivalent embeddings with the same

prime end structure and the same set of accessible points.

(5) The relationship between the prime end structures

of a tree-like indecomposable continuum and the quotient

continuum induced by shrinking each of a collection of

proper subcontinua to a point.

(6) The utilization of the latter to extend theorems of

W. Lewis concerning the pseudo arc, and of J. Krasinkiewicz

and S. Mazurkiewicz concerning accessibility of points,

subcontinua, and composants of indecomposable plane

continue.














CHAPTER 1
INTRODUCTION:
THE FIXED POINT PROBLEM FOR
NONSEPARATING PLANE CONTINUE

1.1. Motivation

A long-standing problem in plane topology is the

following: Do nonseparating plane continue have the fixed

point property for continuous maps? For homeomorphisms?

This question, Problem #107 in the Scottish Problem Book,

credited to Sternbach, has received attention from a number

of mathematicians. A survey of fixed point problems and

theorems, including the above question, has been made by

R.H. Bing [1969].

A number of partial solutions to the problem are known.

Chainable continue, known to be planar [Bing, 1951], have

the fixed point property for continuous maps [Hamilton,

1951]. A closed disk in the plane has the fixed point

property by the well-known Brouwer Fixed Point Theorem.

Early known results also include that nonseparating Peano

continue have the fixed point property [Borsuk, 1932], as do

one dimensional nonseparating arcwise connected tree-like

continue (that is trees, dendrites, and the like) [Borsuk,

1954]. More recently Hagopian showed that nonseparating

arcwise connected continue [1971] and uniquely arcwise

connected plane continue [1979] have the fixed point

property.





2

Of particular interest, however, is the Cartwright

Littlewood-Bell Theorem ([Cartwright and Littlewood, 1951]

and [Bell, 1967] ) showing that nonseparating plane continue

have the fixed point property for extendable homemorphisms.

It has been pointed out by B. Brechner in conversation that

this result extends to essentially extendable

homeomorphisms: those homeomorphisms of a plane continuum

which can be extended to the plane under some reembedding of

the continuum, even if not extendable in the given

embedding. (See Brechner [1981].) This latter result

suggests that questions of embeddings of continue into the

plane may be of some significance in making further progress

on the problem.

In the more general case of continuous maps, an

important result is a consequence of a theorem due

independently to H. Bell [1978] and K. Sieklucki [1968] : a

nonseparating plane continuum admitting a fixed point free

map must have a Lake-of-Wada channel in its boundary for

every embedding into the plane (Embedding corollary 2.5

[Brechner and Mayer, 1981 ].) (For the original Three-Lakes

of-Wada construction see Hocking and Young [1961].) The

choice of topics in this essay is motivated by the

Bell-Sieklucki Theorem and the Embedding Corollary, which

seem to reveal much about the structure of any nonseparating

plane continuum that could admit a fixed point free map.

For instance, if it were the case that every

nonseparating nondegenerate plane continuum with a

Lake-of-Wada channel in its boundary could be reembedded in






3

the plane so that the channel disappeared, then the fixed

point property would follow. In [1980a] B. Brechner and J.

Mayer define the concept of a principal continuum: a

nonseparating continuum that has a Lake-of-Wada channel in

its boundary in every embedding into the plane. Thus one

question of significance to the fixed point problem (asked

in [1980a ]) is: Do there exist principal continue?

1.2 Embeddings and Prime End Structure.

Our primary interest is in embedding problems, and our

primary tool in investigating and comparing embeddings will

be their prime end structures. Prime ends, a concept whose

original development is due to Caratheodory [1913] are a

means of studying and organizing the approaches to the

boundary of a simply connected plane domain. While the

complement of a nonseparating continuum in the plane E is

not simply connected, the complement of such a continuum in
2
the sphere S is simply connected. As we can always insure

that the embedding of a continuum into S misses the point

at infinity, there is no practical difference for our
2 2
purposes in whether embeddings are in E or S Ursell and

Young [1951] have pointed out that prime end theory can be

developed just as well in the latter context.

Through prime end theory we can identify the set of

prime ends of a nonseparating continuum with the set of

points on the boundary of the unit circle B in E (or S ).

For basic concepts and references for prime end theory, we

refer the reader to Brechner [1978]. A sketch of essential

concepts is included herein where needed in the various

chapters. (See 2.2.8 and 4.2.1.)






4

A continuum with a Lake-of-Wada channel in its boundary

must have a particular kind of prime end associated with it

(Theorem 2.9 in Brechner and Mayer [1980a].) The

arrangement of points on the boundary of the unit circle and

their correspondence to kinds of prime ends determine the

prime end structure of a continuum in S Thus we can

detect Lake-of-Wada channels through prime and structure.

Furthermore, it is shown in Brechner and Mayer [1980a]

that if two embeddings of a continuum into S are equivalent

(in the sense that there is some homeomorphism of S onto

itself carrying one embedding onto the other) then the two

embeddings have the same prime end structure (Theorem 2.11

of [1980a].) However, in passing from the embedding itself

to the prime end structure, some information is lost. The

converse of the above theorem fails, as we show in Chapter

4.

1.3 Lake-of-Wada Channels.

In succeeding chapters, a Lake-of-Wada channel running

throughout the entire boundary of a continuum is called a

simple dense canal, a concept due to Sieklucki [1968] and

reiterated in Brechner and Mayer [1980a], Definition 2.1.

The precise definition is given herein in Section 2.2.9.

Roughly, a simple dense canal (s-d-c) is a ray in the

complement that converges to the entire boundary of a

nonseparating continuum in S in such a way that the ray

goes down an ever-narrowing Lake-of-Wada channel in the

boundary. We call an embedding of a continuum with a s-d-c

in the complement a principal embedding.






5

In Chapter 2 we show that chainable continue, if
2
indecomposable, can always be embedded in S with a s-d-c.

(No continuum with decomposable boundary can be embedded

with a s-d-c in the boundary.) Also, if an indecomposable

chainable continuum has at least one endpoint, it can be

embedded without a s-d-c, that is, nonprincipally embedded.

Chapter 2 also includes a wealth of examples of chainable

continue embedded with one or more s-d-c's, including

countably many and uncountably many. Therein and throughout

liberal use is made of figures and diagrams to graphically

represent the continue constructed. For us, the study of

embeddings of plane continue has a significant visual

component, at least on the intuitive level.

In Chapter 3 we present an example of an atriodic

nonchainable continuum in S embedded with a s-d-c. The

example presented is of the type of tree-like plane

continuum most likely to admit a fixed point free map, if

any do. The likelihood is suggested by the properties the

example shares with the tree-like continue admitting fixed

point free maps recently constructed by D. Bellamy [1980].

Bellamy's example is trodic and nonplanar. The

Fugate-Mohler modification of his example is atriodic, but

it is not known if it is planar or if it is nonplanar.

In Chapter 4 we show that both the sin 1/x continuum

and the Knaster U-continuum (or bucket-handle) have

uncountably many inequivalent embeddings into the plane with

the same prime end structure. This shows that an

equivalence of prime end structures, even between






6

homeomorphic continue, cannot be "lifted" to an equivalence

of the embeddings.

In Chapter 5 we investigate the relationship between

the prime end structure of a nonseparating continuum in S2

and the prime end structure of certain quotient continue in
2
S formed by shrinking subcontinua of the given continuum to

points. This enables us to extend certain theorems in

Chapter 2; in particular, we show that no chainable

continuum with an end subcontinuum (not equivalent to

endpoint) is principal. We are also able to extend theorems

of Mazurkiewicz and Krasinkiewicz regarding accessibility of

composants of indecomposable plane continue.

In Chapter 6 we give several examples of triodic
2
continue embedded in S with s-d-c's. Such continue as we

construct exhibit periodic homemorphisms interchanging the

channels. By extending examples of W. Lewis [1981] we show

that certain chainable continue also have this property.

Of the articlesreferenced above, Brechner [1978] and

Brechner and Mayer [1980a] are essential background to the

topics and treatment of this essay. With that proviso, the

following chapters are relatively independent, though all

interconnected by the common themes of the fixed point

problem and of embeddings of plane continue. Essential

definitions, theorems, and concepts are reviewed in each

chapter to the extent that they will be utilized in the

development of that chapter. This results in some

replication of key definitions and important theorems.














CHAPTER 2
EMBEDDINGS AND PRIME END STRUCTURE
OF CHAINABLE CONTINUE

2.1 Introduction.

In this chapter we apply several concepts arising out

of a promising approach to the fixed point problem for

nonseparating plane continue to chainable continue and their

embeddings into the plane. The concepts of principal

embedding and principal continuum were introduced by

Brechner and Mayer [1980a, 1980b]. In the present chapter

they are applied to show two main results:

2.1.1 Theorem.

Every indecomposable chainable continuum can be

principally embedded into E.

2.1.2 Theorem.

No chainable continuum with at least one endpoint is

principal.

It follows from a theorem proved independently by Bell

[1978] and Sieklucki [1968] that any nonseparating continuum

X which admits a fixed point free map contains a minimally

invariant indecomposable subcontinuum X0 in its boundary,

and furthermore, the continuum X0 formed by the union of X0

and its bounded complementary domains, if any, must have the

property that every embedding of it into the plane has a

Lake-of-Wada channel. We call a Lake-of-Wada channel a

simple dense canal (s-d-c). (See 2.2.9 for precise





8

definition.) Such an embedding is principal. (See

Embedding Corollary 2.5 of Brechner and Mayer [1980a].) The

concept of a simple dense canal is due to Sieklucki [1968].

In Section 2, embeddings, both principal and

nonprincipal, of the Knaster U-continuum are discussed.

Lemmas 2.2.5 and 2.2.6 are instrumental in the proofs of

Theorems 2.1.1 and 2.1.2.

In Section 2.3, Theorem 2.1.1 is proved. The principal

embedding of the U-continuum discussed in 2.2.11 is the

model for the principal embedding of any indecomposable

chainable continuum.

In Section 2.4, Theorem 2.1.2 is proved. Techniques

similar to those of Section 2.3 are used, but with the

nonprincipal embedding of the U-continuum discussed in 2.2.7

and 2.2.8 as the model. We also raise several questions

regarding strenghthenings of Theorems 2.1.1 and 2.1.2.

In Section 2.5, examples are given for each n, 1 L n
0o
and n = 2 of a chainable continuum M embedded with
n
exactly n simple dense canals. These continue have exactly

one endpoint and can be reembedded without s-d-c's in

accordance with Theorem 2.1.2. Similar examples of

chainable continue with multiple endpoints and with no

endpoints are given. We make extensive use of

"inverse-limit-with-embedding" diagrams, introduced in

Brechner and Mayer [1980a] and here discussed in 2.2.7, in

presenting these examples.






9

In what follows, all spaces are metric and all continue

are compact connected spaces, and for convenience,

nondegenerate.

2.2 Embeddings Of The Knaster U-Continuum.

Our technique in proving Theorems 2.1.1 and 2.1.2 will

be to show that any indecomposable chainable continuum can

be embedded in the plane as the intersection of a defining

sequence (Dii_ of chains of open disks such that the

pattern that D. follows in Di1_ can be consolidated to the

pattern that one chain of a defining sequence for the

Knaster U-continuum follows in its predecessor. (Since we

shall often refer to sequences indexed by the positive

integers, we shall write them as (Dii., omitting the limits

on the index.) The consolidation can be done in such a way

that the diameter of the links of the consolidations goes to

zero. That some such consolidations of defining sequences

for any indecomposable continuum exist follows from Bellamy

[1981] and Rogers [1970], but for the diameter of the links

to go to zero requires a stronger condition than

indecomposability; chainability suffices. The definitions

of chains, links, and consolidations of chains are standard.

(See Bing, [1948].)

2.2.1 Definition.

A defining sequence (G.]. for a continuum X is a

sequence of open covers of X such that for each i, (1) G. is
1
coherent, (2) Gi+. closure refines G., (3) the mesh of G. is
i
less than 1/2 (4) G. minimally covers X, and (5) X =iQGi.
*
(G. denotes the union of the links of G..) This definition






10

is due to Ingram and Cook [1967], though we have

strengthened condition (3).

2.2.2 Theorem.

A continuum X is indecomposable iff there is a defining

sequence [G.i.i for X such that for each i there is a j>i

such that if G. = L1U L2,L1 and L2 being coherent
*
subcollections of G., then either L1 or L2 meets every

set in G. [Ingram and Cook, 1967].

2.2.3 Definition.

In order to describe an embedding of a chainable

continuum as the intersection of a defining sequence of

chains of open disks in the plane, we make use of the notion

of the pattern that one chain follows in another which it

refines due to Bing [1948] Roughly, a pattern is a map

which tells us in which links of the containing chain the

links of the refining chain sit. Thus for chains C = C(l),

C(2), ... C(m)] and D = [D(1), D(2), ... D(n)], with D

refining C, the pattern D follows in C is a set of ordered

thh
pairs ((xl,,Y), ... (Xnyn ) which indicates that the xi

link of D is a subset of the yith link of C. A pattern is

not necessarily unique, since a link of D may lie in the

intersection of two links of C. Consecutive yi 's may not

differ by more than one. We say a pattern reverses when for

i > 2, from x. to x i+, both yi+l < Y. and yi Yi_-i or both

Yi+ >yi and yi and yi S yi.,. A reversal in a previously
constant or increasing (decreasing) immediately preceding

part of the pattern represents a bend in the chain. Because

patterns are not unique, the same chaining may correspond to






11

both a pattern with and a pattern without a reversal. Such

"pseudo-bends" occur in the intersection of two links of the

containing chain.

2.2.4 Definitions.

A pattern alone does not determine an embedding. In

order to distinguish various embeddings of the same pattern,

we define a pair of notions due to Brechner [1978] and

expand some upon them. Let C = [C(l), ... C(m)) and D =

(D(l), ... D(n)3 be chains of open disks in the plane with

D refining C. By open disk we mean a convex open set with

simple closed curve boundary. We assume chain C has been

"straightened out" by some orientation-preserving

homeomorphism of the plane so that C(l) is on the left and

C(m) is on the right with the nerve of C parallel to the

horizontal axis. Thus the terms "above" and "below" used

below are unambiguous. Suppose D(l) is a subset of C(k).

Then D is descending in C provided that from C(k), D follows

a pattern in C in such a way that each time the pattern

reverses (a bend), the subsequent subchain of D is below the

preceding subchain (the bend is downward); D is ascending in

C provided that from C(k), D follows a pattern in C in such

a way that each time the pattern reverses, the subsequent

subchain of D is above the preceding subchain (the bend is

upward).

If D follows in C a pattern which contains no

reversals, then we say D is straight through C. If D

consists of two straight through subchains, one from (and

contained in) the first (last) link of C to (and contained






12

in) the last (first) link of C, and the other reversing in

the last (first) link of C and returning to the first (last)

link of C, then we say D is two-to-one in C. If D can be

consolidated to a chain following a two-to-one pattern in C,

then we say D is at least two-to-one in C. By extension of

the above, if D consists of n straight through subchains

running from endlink to endlink of C, then we say D is

n-to-one in C. A chain D following a pattern in C which can

be consolidated to an n-to-one pattern is said to be at

least n-to-one in C.

By the term U-like we denote any at least 2 -to-one

pattern or any consolidation of such a pattern to a
2n-to-one pattern. We also call a chainable continuum

U-like if it has a defining sequence of at least 2 -to-one

chain covers. The thrust of Lemmas 2.2.5 and 2.2.6 is that

all indecomposable chainable continue are U-like.

In what follows we uniformly use "n" subscripted by the

index of the chain cover to denote the last link of the

chain. Thus C.i(n.) denotes the last link of chain C..

That open disk chains can be used to produce an

embedding of chainable continuum X into the plane, that is,

guarantee the existence of a homeomorphism of X into the

plane, is a consequence of Theorem 4 of Bing [1951] .

2.2.5 Lemma.

Let X be an indecomposable chainable continuum. Then

there is a defining sequence [Di.i of chain covers for X

such that for each i>l, there exist positive integers

kl





13

(2) D (k2) C D. (ni1), and (3) subchain D.(klk3) is

minimal with respect to properties (1) and (2). Hence, D.

is at least two-to-one in D._I.

Proof. Let (Cj.3 be a defining sequence of chain

covers for X. Our proof proceeds by progressively selecting

subsequences of the C.'s until we have a subsequence which
3
almost meets conditions (1) and (2) above. A certain

sequence of consolidations of the members of this final

subsequence is then shown to meet conditions (1) through

(3).

Select from CiC.3. a subsequence C aa such that for all
3 3 a a
a, the mesh of C is less than 1/2a+l and the closure of any
a
two consecutive links of Ca+l is contained in one link of

C It then follows from Theorem 2.2.2 that for each a,
a
there is a b>a such that Cb consists of two subchains

Cb(l"yb) and Cb(ybnb) such that both subchains meet every

link of C Furthermore, as the C 's are minimal covers, we
a a
may assume that C (yb) is a subset of an endlink of Ca and

that at least one of the subchains contains a link which is

a subset of the other endlink of C.
a
If C (yb) C Ca(n a) then let c = b. Otherwise, apply

Theorem 2.2.2 again to find a c>b such that C consists of
c
two subchains c (l,yc) and C (yc ,nc) such that both

subchains meet every link of Cb (and so every link of Ca)
D a
and C (y ) is a subset of an endlink of C In either case,

we may now find positive integers rc
C (r ,s ,t ) such that both C (r ) and C c(t ) meet C (1) and
c c c c c c c c a
at least one of them is a subset of C (1), and C (s ) is a
subset of Cc(n).
subset of C (n ).
a a





14

By repeatedly following the above procedure, we may

select a subsequence (Ci.i of (C aa such that for all i>l,

there are positive integers u
C. (u.,v.,w.) such that both Ci(ui) and C (wi) meet Cl(1),

at least one of them being a subset of C _l(1), and C.(v.)

is a subset of C. l(ni 1).

We will produce the sequence (Di.i by consolidating

each C. to a chain D. while preserving closure refinement.
1 1

Let D1 = C. Assume (Di-l]1 have been defined. It follows
1* i i=l
from our selection of the mesh of each C that C.(1)U C (2)

lies in a single link of CjI, and so in a single link of

the consolidation of Cj_1, namely DI. Let Dj(1) =

C. (1)U Cj (2), and D (m) = Cj (m+l)' for 2 < m < nj-l. The

mesh of C. guarantees that the mesh of Dj is less than 1/23.

Closure refinement is preserved, since D (1) is a subset of
I

a single link of DjI. Thus [D.i is a defining sequence

for X.

For each i > 1, the triple u., v., w. corresponds to a

triple r.
Di_1 that the original triple had to Ci_1 except that Di(ri)

and Di(ti) are now both subsets of D._ (1). Let k1 be the
i si"
last link of subchain D.(r.,s.) in D_.(1). Let k = s..
1 1 -1 2 1 i
Let k3 be the first link of D (si,ti) in Dil(l). Then [Di3i

i satisfies conditions (1) through (3).

It can then be shown that for each i, Di+1 can be

consolidated to a two-to-one chain in D. Hence, D.i+ is at

least two-to-one in D.. QED






15

Note that the selection of the k.'s in Lemma 2.5 (and
J
in Lemma 2.2.6 to follow) is not unique.

2.2.6 Lemma.

Let X be an indecomposable chainable continuum. Then

there is a defining sequence Di3i of chain covers for X

such that for each i>l, and for each positive even integer
i i k+i_ uhta
m, there exist positive integers k 1 2 M+l
(1) Di(k')UDi(k') U ... U Di(k'+) C Di(l), (2) Di(ki)U
1 1 3 i mll i2
Di(k )U ... U D m(kl) C Di(ni), and (3) subchain
Di ( i
D (kk +) is minimal with respect to properties (1) and

(2). Hence, D. is at least m-to-one in Dil, and

furthermore, X is U-like.

Proof. Let (Di]. be a defining sequence of chain

covers for X satisfying the conditions of Lemma 2.2.5. Then
for all j > 1, D. is at least two-to-one in D -. Let p be

chosen so that 2p- < m < 2P. Let CDi}i be the subsequence
th
of CDI. chosen by selecting the first, the p+1 and at

the nth stage, the (n-l)p+lth member of CD.]j.. Then for

each i > 1, D. will be at least 2P-to-one in D. i. But as

2P is no less than m, each D. is at least m-to-one in D. i.

That CD.]i satisfies conditions (1) through (3) can then be

shown. QED

A version of Lemma 2.2.6 can be proved for odd m,

though with ki for odd j in D i(1) and k' for even j in
3 3
i
D (n._), 1 < j < m+l. That is, D (km+) lies in the last
link of D not the first.l
link of D il'1 not the first.






16

2.2.7 Standard Embedding of the Knaster U-continuum.

We can describe what we will call the standard

embedding of the Knaster U-continuum by a defining sequence

[Ci}i of open disks in the plane such that C i+1 follows in

C. a descending two-to-one pattern of embedding. This
1

two-to-one pattern and descending embedding is schematically

represented by the "inverse-limit-with-embedding" diagram in

Figure 2.1. The infinite repetition of the indicated

pattern-with-embedding produces the standard embedding of

the U-continuum. We suppose C. to be straightened out to
1

guide us in embedding C i+1 in Ci. A point p of a chainable

continuum X is an endpoint if, and only if, for some

defining sequence for X, p is the intersection of first

links. Note that p = .Ql1Ci(1) is the only endpoint of the

U-continuum.

2.2.8 Prime End Structure Of The Standard Embedding Of The

Knaster U-Continuum.

Basic definitions concerning prime ends and further

references may be found in Brechner [1978] Figure 2.2.

represents the only accessible composant of the standard

embedding of the U-continuum. A composant is accessible if

it contains at least one accessible point. In this case the

composant is accessible at every point, by an argument

similar to the proof of Lemma 2.4.1.1. Note (Qi.i is a

chain of crosscuts of E -U such that Q.- p. This chain of

crosscuts defines a prime end E. By I(E) we denote the

impression of E, the intersection of the closures of the

bounded domains of Q.iU U, over all i. (These domains are






17

towered.) It can be shown that I(E)=U, and for any prime

end F distinct from E, such as that defined by [Ri.i in

Figure 2.2, I(F) is a single point. The existence of at

least one prime end E with I(E)=U is necessitated by U's

indecomposability [Brechner, 1978] .

By P(E) we denote the set of principal points of prime

end E, the set of point to each of which some chain of

crosscuts defining E converges. It can be shown that

P(E)=[p], the endpoint of U, for E defined by [Qi.i above.

Such a prime end is of the second kind: I(E) nondegenerate

and P(E) degenerate. Any other prime and F of U is of the

first kind: I(F)=P(F), both degenerate. For any prime end

E, each of I(E) and P(E) is a continuum (or degenerate) with

P(E) a subcontinuum of I(E).

2.2.9 Definition

A simple dense canal (s-d-c) of the continuum X is a

ray D C E2-X (a 1-1 continuous image of [0,- )) such that

(1) D-D=Bd X, (2) at each point of D there is a transverse

crosscut to X, and (3) diameter of such crosscuts goes to

zero as we go to infinity on D. A crosscut Q is transverse

to D at a point d if, and only if, QfnD = (d3, and for a

sufficiently small open disk neighborhood U of D, each

component of U-Q contains exactly one component of (Un D)-

d.

2.2.10 Theorem.

A nonseparating plane continuum X has a simple dense
2
canal iffthere exists a prime end E of E -X such that I(E) =

Bd X = P(E). (Theorem 2.9, [Brechner and Mayer, 1980a].)







2.2.11 Definitions.

A prime end E such that I(E) = P(E), both

nondegenerate, is of the third kind. So a s-d-c corresponds

to a prime end of the third kind for which I(E) = Bd X. For

tree-like continue, the necessary and sufficient condition

reduces to I(E) = X = P(E). Such embeddings of

nonseparating continue are termed principal embeddings. If

all the embeddings of X into the plane are principal, then X

is called a principal continuum ( [Brechner and Mayer,

1980a], Definition 2.6). In virtue of the standard

embedding, the Knaster U-continuum is not a principal

continuum.

2.2.12 Principal Embeddings of the U-Continuum.

In Remark 3.5 of Brechner and Mayer [1980a], we

indicated that the U-continuum has a principal embedding.

Consider the defining sequence for the U-continuum formed by

taking each odd-numbered chain of the standard defining

sequence. Let that be [Ci. Note that the pattern one

odd-numbered chain of the standard sequence follows in

another is four-to-one. Embed the U-continuum according to

the inverse-limit-with-embedding diagram of Figure 2.3(a)

repeated infinitely. The hatched line in Figure 2.3(a)

represents an initial arc of a ray D extendable in such a

way as to be dense in U. Transverse crosscuts to U exist at

each point of D, and as they can be confined to a single

link, their diameter goes to zero. Hence D is a simple

dense canal in this embedding of U. Note also that any

prime end not corresponding to D is such that its impression






19

is degenerate. Furthermore, in this embedding endpoint p

and its composant are inaccessible. Figure 2.3(b)

illustrates another principal embedding of the U-continuum,

but with the endpoint accessible. This embedding has two

prime ends whose impression is all of U, but only one of

them corresponds to a s-d-c. An endcut to p corresponds to

the other prime end.

It should be noted that the embeddings represented by

Figures 2.1, 2.3(a), and 2.3(b) of the U-continuum are

inequivalent. This is evident in comparing the differing

accessibility of endpoint p in Figures 2.1 and 2.3(a). That

2.3(b) is also not equivalently embedded with respect to

either of the other is a consequence of the differing prime

end structures. (See Theorem 2.11 of Brechner and Mayer

[1980a].)

2.3 Principal Embeddings Of Chainable Continua.

We now have sufficient tools to show that any

indecomposable chainable continuum can be principally

embedded. We will use the pattern to Figure 2.3(a) to

produce an infolded U-like embedding of any indecomposable

chainable continuum.

2.3.1 Proof Of Theorem 2.1.1.

Let X be an indecomposable chainable continuum. By

Theorem 4 of Bing [1951], X can be realized as the

intersection of a defining sequence of chains of open disks

in the plane. However, we want a particular kind of

embedding, one with a s-d-c in the complement.






20

There is a defining sequence C.3.ii for X satisfying the

following conditions:

(1) C. is taut: that is, any pair of nonadjacent links
1

of C. is a positive distance apart.
1

(2) If a link of C. (other than an endlink) lies

entirely in Ci1 (k), for any k, 1 < k < ni_,11 then that
link is contained in a subchain of C. of nine links
1

lying entirely in C_ (k).

(3) C. is at least four-to-one in Cil; that is, for

each i > 1, there exist positive integers r. < s. < ti

< u. < v. such that:
1 1

(a) Ci(ri)U Ci (ti)UCi (vi) C Ci1(l),

(b) Ci(si)U Ci(ui) C Cil(nil),

(c) subchain C. (ri,v.) is minimal.

Conditions (1) and (2) follow from the fact that X is

compact. Condition (3) follows from Lemma 2.2.6.

We may think of the C.-covers abstractly. Then X can

be embedded in the plane by specifying a particular defining

sequence D.i.i of chains of open disks in the plane whose

intersection can be shown to be homeomorphic to X. Such a

homeomorphism is defined in Theorem 11 of Bing [1948],

which Bing uses to prove Theorem 4 of 1951 1. What is

required to apply Theorem 11 is that D. follow in D i_- the

pattern that C. follows in C. I.

In producing the embedding below, we think of the

D.-chains semi-abstractly as chains of topological open
1

disks with as yet no particular embedding in the plane. We

then specify how these open disks are to be situated in the





21

plane to produce the desired embedding of X. Figure 2.4(a)

is an example of the pattern that some D. might follow in

D i_. For convenience, we have written the pattern

descending, but at the moment think of it abstractly, with

no particular embedding.

In carrying out our embedding, we make use of an

auxiliary sequence of open disk chains [Ui)i, such that Ui

refines D il exactly four-to-one and contains D. in a

particular way. Chain U.(l,m.) consists of four straight

through subchains Ui(l,hi), Ui(hi,ji), Ui(Jilki) and

Ui(ki,mi) with 1 < h. i < ji < ki < mi, and links Ui(l),

Ui (ji), and Ui(mi) lie in Di_1(l), while links Ui(hi) and

Ui(ki) lie in Di l(nil). Furthermore, Ui(l,hi) contains

D. (l,ri,si), Ui(hi,ji) contains Di(si,ti), Ui(ji,ki)

contains D. (t.,u.), and Ui(k.,m.) contains Di(ui,v.,ni).

The links of D. numbered r., s., ti, u., v. lie in the links

of U. numbered 1, hi, ji, ki, mi' respectively. In placing

a link of D. in a link of U., we preserve the pattern that

D. follows in D I. That is, if Di (a) C Ui (b) and Ui (b)C

Di-l(c), then D (a) C D -l(c) in the original pattern. Note

that D.(1) may be in any link of U.(l,h.), and Di(n.) may be

in any link of Ui(ki,mi), depending upon what the original

pattern was.

Suppose that Di_ has been realized as a chain of open

disks in the plane. Then U. is embedded in D i_ with the

infolded pattern-with-embedding of Figure 2.4(b). We have

placed no restriction on the mesh of U. other than that U.

refine Di_. There could be as few as four links of U. (one






22

from each subchain) in an interior link of D.i, essentially

subdividing that interior link into four links, and as few

as two links of U. in D i-(ni-), just Ui(hi) and Ui(ki).

However, in order to keep the links of U. open disks, we

need at least five links in D i_(1): Ui(l), Ui(mi), and a

three-link subchain with U. (j.) in the middle. Such a

U.-chain meeting these minimal requirements is illustrated
1

in Figure 2.5.

We now embed subchains D.(r.,si), D.(si,ti), Di(tiui),

and D.(u.,v.) descending in Ui(l,hi), Ui(hi,ji), Ui(Ji,ki),

and U.(k.,m.), respectively. This pattern-with embedding is

shown for our example in Figure 2.4(c). Compare also Figure

2.5. Note that links of D. numbered r.i, s.i, t.i, u.i, v. lie

in links of U. numbered 1, hi, ji, ki, mi, respectively, and

that the pattern D. follows in D._1 is preserved, as can be

seen by comparing Figures 2.4(a) and 2.4(d). Since there

are at least nine links of D. in the subchain lying in
1

D il(1) that contains D.(t.), we can require that Di(ti)

have four links of D. both preceding and following it lying
1

in D. i(1). Hence there are enough links for D. to refine

U.i(j.-1, j.+l) C Dil(1), and still have the diameter of

links of D. less than half the diameter of links of Di.1

The initial and final subchains of D. are embedded with
1

D.(l,r.) descending U.(l,h.) and above D (r.,s.) in
1 1 1 1 1-- 1l 1
Ui(l,hi), and Di(vini) ascending in U (ki,mi) and above

D.(u.,v.) in U.(k.,m.). The resulting pattern-with-

embedding diagram for our examples, showing how D. now sits

in D. i, is illustrated in Figure 2.4(d). How the chains






23

D i_ (straightened out), U., and D. might appear in the

plane is shown in Figure 2.5.

A s-d-c D may then be constructed stage-by-stage as we

illustrated with the principal embedding of the U-continuum.

An arc of the canal in our example is shown as the hatched

line in both Figures 2.4(d) and 2.5. That D.i(n.) may not

lie in D i_(1), in contrast to the situation with the

U-continuum, can only make this arc of the canal

proportionately longer. QED

2.4 Nonprincipal Embeddings Of Chainable Continua.

For a decomposable chainable continuum, every embedding

into the plane is nonprincipal, as a s-d-c in X implies Bd X

is indecomposable ( [Brechner and Mayer, 1980a] Theorem 2.9.)

For an indecomposable chainable continuum, we can get an

embedding using the two-to-one standard, rather than the

infolded four-to-one, embedding pattern for the U-continuum

as our model. However, while the standard embedding of the

U-continuum has no s-d-c, we have as yet been able to

produce such an embedding (without a s-d-c) of an

indecomposable chainable continuum, in general, when the

continuum has at least one endpoint. Various examples (see

our 2.5.7) of chainable continue without endpoints have

proved reembeddable without a s-d-c, but the general proof

eludes us, though we conjecture that it can be done.

2.4.1 Proof Of Theorem 2.1.2

Let X be an indecomposable chainable continuum with

endpoint p. We can find a defining sequence (Ci]i of chain

covers of X satisfying the following conditions:






24

(1) p is in Ci (1)-Ci (2); hence Ci (1) C Cil(1).

(2) Ci is taut. (As defined in 3.1(1).)

(3) If a link of C. (other than an endlink) lies
1
entirely in Ci_-1(k), for any k, 1 < k n -i, then that

link is contained in a subchain of C. of nine links

lying entirely in C. i(k).

(4) C. is at least two-to-one in Cil; that is, for

each i > 1, there exist positive integers r. < S. < t.

such that:
(a) Ci (ri)UCi (ti) C C i(1),

(b) Ci (si) C Cil (ni1),

(c) subchain C. (r.,s.,t.) is minimal.

(5) Subchain C.(l,r.) is minimal; that is, no link of

Ci(l,ri) lies in Ci1 (ni.1).

Condition (1) holds because p is an endpoint of X. See

Section 5 of Bing [1951]. Conditions (2) and (3) follow

from the fact that X is compact. Condition (4) is a

consequence of Lemma 2.2.5. Since first links are towered,

we may choose the first subchain satisfying condition (4),

thus satisfying condition (5).

As in the proof of Theorem 2.2.1, our procedure for

embedding X is to specify a defining sequence [D.i) of

chains of open disks in the plane such that D. follows in
1
Di_1 the pattern that C. follows in C i. We think of (D.i.i

semi-abstractly, as chains of topological open disks with no

particular embedding. Figure 2.6(a) indicates how some D.
1
might sit in Di -, written descending for convenience. In

some cases it could be that r.=l, though not in the example





25

we illustrate in Figure 2.6(a). Also note that s. is not
1
uniquely determined: any of the three "loop ends" of D.
1
that lie in Di l(n _I) in Figure 2.6(a) could contain

Di (si).

In carrying out our embedding we make use of an

auxiliary sequence U.i.i of chains of open disks, such that

U. refines Di_1 exactly two-to-one and contains D. in a

particular way. Chain U.i(l,m.) consists of two straight

through subchains U.(l,k.) and U.(k.,m.) with 1 < k. < mi,

and links Ui(l) and Ui(mi) lie in Di_-l(1), while link Ui(k.)

lies in D -l(n. il). Furthermore, U. (l,k.) contains

D. (l,r.,s.) and U. (k.,m.) contains D. (s.,t.,n.). The links

of D. numbered 1, r., s., t. lie in the links of U. numbered

1, 1, k., mi., respectively. In placing a link of D. in a

link of Ui, we preserve the pattern that Di follows in D. .

That is, if Di (a) C U. (b) and U. (b) C D. -l(c), then D. (a) C

Dil(c) in the original pattern. Note that D.i(1) may be in

any link of Ui(l,hi), and Di(ni) may be in any link of

U.(k.,m.), depending upon what the original pattern was.

Suppose that Di_1 has been realized as a chain of open

disks in the plane. We embed U. in Dil following the

descending pattern-with-embedding of Figure 2.6(b). We then

embed subchains D.(l,r.,s.) and D.(s.,t.) descending in

Ui(l,k.) and Ui (ki.,mi), respectively. This pattern-with-

embedding is shown for our example in Figure 2.6(c). Note

that links of D. numbered 1, ri, s., t. lie in links of U.

numbered 1, 1, ki, mi., respectively, and that that pattern

Di follows in Di_ is preserved. We embed the final






26

subchain D.(t.,n.) in U.(k.,m.) descending and below

D.(s.,t.) in U. (k.,m.). Note that D. is descending

throughout U.. Figure 6(d) illustrates the resulting
1
embedding of D. in Di_. Note that D (l,s.) is descending

in Di_1 and that the initial subchain of D i+ will be

descending in D.(l,si).

Figure 2.7 represents the first three stages of a

typical continuum X embedded according to the above

procedure. The intervening U. chains have been omitted.
1
The heavier line is the minimal subchain D3(r3,s3,t3).

It is the embedding of X constructed above that we

claim has no simple dense canal. We prove this claim by

showing that only one prime end E, defined by a chain of

crosscuts converging to endpoint p = i1D.i (1) = in1Ui(1), is

such that I(E) = X. But, as p is accessible with respect to

E, P(E) = (p), so E is not of the third kind, so does not

correspond to a s-d-c. We will show that every prime end F

distinct from E is such that I(F) is a proper subcontinuum

of X, so even if F is of the third kind, it does not

correspond to a s-d-c. (It is possible for a point, even an

endpoint, to be accessible with respect to one prime end and

also be a principal point of a distinct prime end, including

a prime end corresponding to a s-d-c, as is illustrated in

Figure 2.3(b) for a principal embedding of the U-continuum.

Example 2.5.1 also has this property. Hence we cannot

simply assume that a chain of crosscuts defining prime end F

does not converge to p.)






27

Our proof is similar to those of Theorems 4.3.1 through

4.3.3 of Brechner [1978]. It will be convenient and aid

comparison to divide our proof into three lemmas roughly

corresponding to Brechner's theorems.

2.4.1.1 Lemma. All accessible points of X are in the

composant C of endpoint p.
p
Proof. Assume D1 is straightened out and let R and S

be rays in the plane drawn respectively upward and downward

from endpoint p of X so that R (respectively, S) meets

Bd D.i(1) in exactly one point and otherwise meets no other

link of DI. Then RUS separates E into right and left

half-planes with X contained in the closure of the right

half-plane, and only p in X in the closure of the left

half-plane. (See Figure 2.7.)

Let q be any accessible point of X distinct from p.

There are two homotopy classes of crosscuts in the right

half-plane minus X from p to q: those like T1 in Figure 2.7

which go around X from p to q, and those like T2 in Figure

2.7 which do not go around X. Let A be a crosscut from p to

q which, for some k, is homotopic with fixed endpoints in

the right half-plane minus X to a crosscut which, except for

short terminal segments in Dk(l) and the link of Dk

containing q, lies entirely above Dk in the right half-plane

when Dk is straightened out. We can do this because A can

be chosen so that its image under a straightening-out

homeomorphism lies above some straightened out Dk chain. We

may assume that for all i, A meets only those links of D.
1that contain p and q.
that contain p and q.






28

Let K be the subcontinuum of X irreducible between p

and q. Let C. be the subchain of D. which minimally covers
1 1
K. The first links of (Ci.i are towered on p. Since A

enters a link of Dk from above, it meets a link of

Dk+l (lSk+l). But Dk+l (1,Sk+l) is descending in Dk and no

link of Dk+l lies above D k+l(,1sk+l ). All subsequent

initial subchains of CDi]i=k+2 that lie in Dk+l(l,sk+l ) are

descending in their immediately containing chain, and at

each stage, no subsequent subchain lies above them. Hence,

the last links of CC.i.i are towered on q. Though C.(1) is

in every case the first link of Di, the last link of C. is

not the last link of D. for all i > k. Indeed, iQC.i will

be at least some fixed distance d from any point of X in

D k(n ). So K = nC.#X. Hence, p and q lie in a proper
k+l k+l i=l i
subcontinuum of X, and so are in the same composant.

QED (2.4.1.1)

Endpoint p is accessible, and therefore, a prime end E

corresponding to an endcut to p is defined by any chain of

crosscuts (Q.]). which cuts R and S in exactly one point each

(or is homotopic to such a chain). Any such chain converges

to p, so p is the only principal point of E. In Figure 2.7,

Q1 and Q2 illustrate such a chain. Since E has only a

single principal point, E does not correspond to a s-d-c.

2.4.1.2 Lemma. If F is a prime end of X distinct from

E, then I(F) is a proper subcontinuum of X (indeed, of C .)

Proof. Let F be a prime end of X distinct from E and

[Pi]i a chain of crosscuts defining F. If some subchain

(P.]. of [P.i. is such that P. intersects R and S exactly
11 J





29

once each (or is homotopic with fixed endpoints in E -X, to

such a crosscut), then (P.]. defines prime end E. So there

is a J, such that for all i > J, P. is homotopic with fixed

endpoints in E -X, to a crosscut that does not intersect RU

S. We may assume that J=l, and that (P.i.i is a chain of

crosscuts, none of which intersect RUS. In Figure 2.7, P1

and P2 illustrate such a chain of crosscuts.

P1U X separates the right half-plane of E and the

bounded domain cut off by P U X contains P. for all i > 1.

Because of the U-like embedding, we may straighten out

chains without disturbing RUS, so that P1 may be seen to be

homotopic to a crosscut which lies above Uk, and therefore

above Dk, in the right half-plane, for some k. We may

assume P1 meets only those links of U. and D. that contain

the endpoints of P1. Then, as in the proof that X contains

only one accessible composant, the continuum irreducible

between the endpoints of P1 is a subcontinuum of the

accessible composant C of X, and so is properly contained

in X. Since the domains cut off by a chain of crosscuts are

towered, I(F) is properly contained in X. QED(2.4.1.2)

In Figure 2.7, PI and P2 are the first two crosscuts in

a chain defining some prime end F distinct from E. Note

that if the bending of subsequent chains Di., for i > 1, is

back toward endpoint p and descending, as in D2 and D3, then

the continuum irreducible between the endpoints of P1 may

contain endpoint p. This is the case if X is the pseudo arc

[Brechner, 1978].






30

2.4.1.3 Lemma. There is only one prime end E of X such

that I(E) = X, but X contains no s-d-c.

Proof. From Lemma 2.4.1.2 it follows that no prime end

F distinct from E corresponds to a s-d-c. We have already

observed that E does not correspond to a s-d-c. As X is

indecomposable, some prime end must be such that its

impression is X. Only E qualifies, so I(E) = X, but X

contains no s-d-c. QED(2.4.1.3)

With Lemma 2.4.1.3 we conclude the proof of Theorem

2.1.2, having shown that every chainable continuum with at
2
least one endpoint can be nonprincipally embedded in E It

would be welcome if the condition that the continuum have an

endpoint could be eliminated.

2.4.2 Question.

Can every chainable continuum be nonprincipally

embedded?

If so, then the fixed point property for chainable

continue would follow as a corollary to Embedding Corollary

2.5 of Brechner and Mayer [1980a].

2.4.3 Question.

Can every indecomposable tree-like plane continuum be

principally embedded?

2.4.4 Question.

Can every tree-like plane continuum be nonprincipally

embedded?

If so, then all tree-like plane continue have the fixed

point property. We conjecture, though we have no proof,

that the atriodic nonchainable continuum in Mayer, [1980b]






31

and in Chapter 3 is a principal continuum. If our conjecture

is true, then the answer to 2.4.4 is no.

2.5 N-Principal Embeddings Of Chainable Continua.

An embedding of continuum X into the plane with exactly

n s-d-c's is termed an n-principal embedding. If every

embedding of X into the plane contains at least n s-d-c's,

then X is an n-principal continuum ([Brechner and Mayer,

1980a], Definition 2.6.) In this section we present

examples of chainable continue with n-principal embeddings.

Details of the proofs that our examples have the

properties we mention are left to the reader. Figures are

provided for the most part, and results stated are generally

evident from the figures and the techniques used in the

proofs of Theorems 2.1.1 and 2.1.2.

2.5.2 Example

A chainable continuum M can be constructed for each
n
n > 0 embedded with exactly n s-d-c's. The Knaster

U-continuum in an embedding in which two-to-one chains are

alternately ascending and descending is M0. For each n > 0,

M is defined by a sequence of chains (of open disks in the
n
plane), each member of which consists of 2(n+l) straight

through subchains (though not all straight through the

entire containing chain). A pair of consecutive subchains

comprises a loop, which consists of a straight through chain

from the first link of the containing chain to a designated

link, then returning straight through to the first link.

There is one "longest" loop which is straight through from

end to end of the containing chain, succeeded by n "shorter"






32

loops, of graduated "length." The embedding alternates

ascending and descending versions of the above pattern.

Figure 2.8(a) illustrates the inverse-limit-with-embedding

diagram for M2. Note the alternation of an ascending with a

descending pattern. The infinite repetition of this pair of

embedding patterns produces M2. Note that the bends of C3

occur in bend or end links of C2 that lie in C1(1). It can

be shown that M2 (and Mn for any n) has exactly one

endpoint which is the limit of all bend and end links. To

show that there is just one endpoint, show that every other

point is an interior point of an arc; that is, is in an

interior link at every stage of a defining subsequence of

straight through subchains.

Figure 2.8(c) illustrates several stages in the chain

construction of M2 in the plane. There, and in Figure

2.8(a), the lines marked with differing symbols represent

the two s-d-c's. Note that a segment of a s-d-c passing

through links of C2 and between upper and lower subchains of

C3 meets every link of C1. In this fashion one can show the

canal is dense in M2. Figure 2.8(b) shows how to reembed M2

with no s-d-c's in accordance with Theorem 2.1.2.

In the 2-principal embedding of M2 (and the n-principal

embedding of any M ) the endpoint is the only accessible

point of its composant. This is because the chains are

alternately ascending and descending, and so "block" endcuts

to other points of the endpoint composant. The endpoint is

also a principal point of every prime end corresponding to

in Figure 2.10 of a chainable continuum M embedded with c
c






33

one of the s-d-c's. Note that an endcut to the endpoint

also corresponds to a prime end whose impression is all of

M2

2.5.2 Example.

We can modify the construction in Example 2.5.1 to

produce a chainable continuum Mo, embedded with a countable

infinity of s-d-c's. At each stage in the sequence of

chains defining Mo,.there is one more loop than in the

immediately preceding stage. Figure 2.9 illustrates the

inverse-limit-with-embedding diagram for the U.-principal

embedding. Note that chains are alternately ascending and

descending. In accordance with Theorem 2.1.2, M,.can be

reembedded with no s-d-c's. The required reembedding is

similar to that of M2 in Figures 2.8(b).

2.5.3 Example.

A chainable continuum with c = 2 s-d-c's can be

constructed in a fashion similar to that of Example 2.5.2 by

adding at each stage an additional loop for every loop of

the preceding stage. We illustrate a different construction

s-d-c's. In general, chain Ci contains 2i-l +1 loops, as

illustrated. Chains are uniformly descending, but alternate

starting in link 1, link n. i+, link n. i+2, link 1, every four

stages. Only three of the s-d-c's are illustrated by the

lines marked with differing symbols in Figure 2.10. That

the number of s-d-c's is 20can be seen by noting that they

can be represented as the branches of an infinite binary

tree. By reembedding Mc with the longest loop on the

outside at each stage, all s-d-c's are eliminated. This






34

particular reembedding does not follow the proof of Theorem

2.1.2.

2.5.4 Example.

The special embedding P of the pseudo arc is produced

by alternating descending with ascending crooked chains

between opposite endpoints p and q of P In [1978]
s
Brechner describes this embedding and conjectures that each

accessible point of P lies in a different composant. In

[1980] Lewis shows this conjecture to be true. The

endpoints of any crosscut of P thus lie in different
s
composants. Hence the continuum irreducible between the

endpoints is Ps It follows that for any prime end E of P s

I(E)=P .

2.5.4.1 Theorem. The special embedding P of the
s
pseudo arc contains uncountably many s-d-c's.

Proof. Let (Ci.i be a defining sequence of crooked,

alternately ascending and descending, open disk chain covers

for the special embedding P of the pseudo arc. We show

that a binary tree can be constructed dense in Ps Each

branch corresponds to a s-d-c. We first show that a ray can

be constructed dense in P and forming a s-d-c, and then

show that countably many branch points can be introduced to

the ray, turning it into an infinite tree.

Suppose ray R enters link C.i(a.) of chain C.. Without

loss of generality we assume R enters C. from above (viewing

Ci as straightened out) and that C. is descending in Ci..

Since C i+l is crooked in C., there is a (crooked) loop

extending from Ci(l) to Ci(ai) and back to C.(2), and the






35

loop is followed by a subchain returning to Ci(ai). Extend

R through links C.i(2,a.) (R is moving in reverse order)

passing between the lower part of the loop and the returning

subchain, and entering a link C i+1 (a i+) sitting in C.i(2).

This situation, with only a few of the bends of C i+ shown,

is diagramed in Figure 2.11(a). Now C i+2 is ascending in

C i+. In link C i+l(a i+) we can find a loop-end, where the

loop of Ci+2 extends from C i+l(n. i+) to C i+l(a i+) and back

to C i+l(n i+-l), and the loop is followed by a subchain

returning to C i+l(ai+ ) (all in reverse order). We can

extend R through links C i+l(ai+ ,ni+ -1) between links of

the lower part of the loop and the returning subchain, and

entering a link C i+2(a i+2) that lies in C i+l(n --1). Note

that R reaches within twice the mesh of C of p and within
1
twice the mesh of C i+1 of q. Proceeding in this fashion, we

can extend R alternately toward p and q, and so that the

closure of R includes P We have constructed R so that
s
crosscuts of decreasing diameter exist at every point, as we

go to infinity on R; hence R constitutes a s-d-c in P .
s
To see that there are uncountably many s-d-c's in P

we modify our construction above. Assume that above we pick

the first loop we come to that fits the described

conditions. (For descending chains this will be the first

loop satisfying the conditions in the order on the chain;

for ascending chains it will be the last loop in that

order.) Note that in link C i+l (ai+l -1) and below the loop

ending in Ci+l1(ai+.) there is a loop, possibly consisting of

our previously designated returning subchain and a subchain






36

from C i+l(a.i+1-1) to C i+l(n. i+-l), with still another

returning subchain back to C i+l(a i+-l) below it. We may

introduce a branch point to R in link C i+l(a i+l) and extend

a branch R' from R at that point to C i+l (a i+l--1) and then

further extend R' through links Ci1 (ai+ -1 ni+1-l) between

the loop and the returning subchain and entering a link

C i+2(bi+2 ) that sits in C i+l(n i+-l). This link of Ci+2

will be distinct from that which R entered, and R and R'

follow separate "channels" between links of C i+2.* We may

repeat this branching maneuver in both C i+2(a i+2) and

Ci+2(b i+2). Figures 2.11(b) and (c) indicate one branching

and the continuation of the rays in separate channels

thereafter. Figure 2.11(d) illustrates an entire crooked

refinement (with minimal bending). Each link (3 to n.-l) of
1
the containing chain is the entrance to a distinct channel,

from top (the hatched lines) and bottom (where we show an

example of branching). By repeatedly branching, we can

extend R so as to form a binary tree dense in Ps so Ps has

uncountably many s-d-c's. QED

2.5.5 Example.

Theorem 3.1 of Brechner and Mayer [1980a] shows that the

three-point continuum has embeddings both with and without

s-d-c's. Continue with multiple s-d-c's can be produced

with exactly two endpoints, with uncountably many endpoints

(the pseudo arc), or with a number of endpoints proportional

to the number of s-d-c's. We describe one series of

examples from the many possible below.






37

The Knaster S-continuum (like the U-continuum in

Example 2.5.1) can be used as a basis for constructing

continue with multiple s-d-c's. Figure 2.12(a) is the

standard embedding of the S-continuum. It has exactly two

endpoints, whose composants are accessible at every point,

and no s-d-c. Figure 2.12(b) is an embedding of the

S-continuum with one s-d-c. (Two embedding patterns

alternate.) We can produce a continuum Z embedded with
n
exactly 2n s-d-c's and having exactly two endpoints. The

embedding of the S-continuum with alternate chains ascending

and descending is Z For Z we introduce n extra loops of

graduated length in both the first and last of the three

straight through subchains defining Z0 in a manner similar

to that of Example 2.5.1. The endpoints of Z are opposite

endpoints, since there is a definnig sequence for Z such
n
that each chain is a chain from one endpoint to the other.

For a continuum with a pair of opposite endpoints it is

enough to embed it with all chains between the two endpoints

descending to eliminate all s-d-c's. The proof is similar

to that of Theorem 2.1.2; show that all but the prime ends

corresponding to endcuts to the opposite endpoints have

properly contained impressions.

To produce continue with n s-d-c's and two endpoints

for odd n, we need only eliminate one of the loops from each

stage of the construction of Zk, k = (n+l)/2. Other

variations on the S-continuum are also possible.








2.5.6 Example.

Bellamy [1980] has constructed an example of an

indecomposable chainable continuum with no endpoints.

Figure 2.13(a) is an example of such a continuum N

discovered independently by us. Continuum N has no s-d-c in

the given embedding, though it does have a simple canal,

dense in a proper subcontinuum of N. Figure 2.13(c)

represents several stages in the chain construction of N in

the plane. The simple canal, not dense in N, is illustrated

by the hatched line in both Figures 2.13(a) and (c). That N

has no endpoints follows from the fact that every point of N

is an interior point of an arc, that is, is in the

intersection of a tower of straight through subchains, and

always in an interior link. The proper subcontinuum N of N

in which the indicated canal is dense is the intersection of

the subchains C.i(l,p.) which are towered. Inspection

reveals N to be homeomorphic to the U-continuum with
u

p =.Ql1Ci(pi) as its endpoint. (Note that Ci(p.) is the link

in which Ci+l (1) and Ci+l (pi+) sit.) Though the canal in N

is dense in Nu, it is not a s-d-c of Nu because there are

not transverse crosscuts at each point to N .

One composant of N, inaccessible in the given

embedding, consists of the union of N with a ray (a 1-1
u

continuous image of [0,-)) at their common endpoint p. If

N is shrunk out of N, the quotient space is the
u

U-continuum. The endpoint of the U-continuum corresponds to

the nondegenerate element of the decomposition. (These

properties of N were pointed out by Beverly Brechner in

discussion.)






39

Figure 2.13(b) illustrates a modified construction

producing continuum N' in which the hatched line represents

a canal that is dense in N'. Note that link C. (p.) now lies

in the second straight through subchain. Note that the

C. (l,p.) subchains are not towered in the sequence defining

N', and indeed, can be used to define an increasing tower of

proper subcontinua whose limit is N'.

2.5.7 Example.

Continuum N, like the U- and S-continua, can be used as

a basis for constructing continue embedded with multiple

s-d-c's. For each n > 0, we can construct a continuum N
-- n
embedded with exactly n s-d-c's and having no endpoints.

Embedding N with alternate chains ascending and descending

produces N0. Figure 2.14(a) illustrates N2.

Compare Figures 2.14(a) and 2.8(a) for the similarity in the

constructions of M2 and N2* Figure 2.14(b) shows how N2 may

be reembedded without a s-d-c. An endcut to p =.Ql1C. i(pi)

corresponds to the only prime end whose impression is N2.

Any of the continue N can be similarly reembedded so as to

have no s-d-c.

As is the case with N = No, each continuum N contains

a composant which is the union of a U-continuum subcontinuum

N and a ray at their common endpoint p. If N is shrunk

out of Nn the quotient space is continuum Mn of Example

2.5.1. Note that in the nonprincipal embedding of N2 given

by Figure 2.14(b), the composant containing N is

accessible, but N itself is accessible only at its endpoint

p. In the 2-principal embedding of N2 given by Figure






40

2.14(a), the composant containing N is again accessible,
u

and now N is accessible, though not at every point (Nu is

indecomposable), and in particular, not at point p.

In the manner of Examples 2.5.2 and 2.5.3 we can also

construct chainable continue N oand N with wo and c s-d-c's,
^o c
respectively.

Any of our examples can be constructed with crooked

rather than straight through chains, thus producing

embeddings of the pseudo arc with n s-d-c's, for each n > 0.





41











C2




Figure 2.1


Figure 2.2






42









o1 1 n1

cD

U2



(a)









C1 1 n1






no2


(b)


Figure 2.3










DiSi



D^ I .
tiK
1Ui

---- u i


13~

ri


1 hi i ki m


ni
(a)







1 ni-i


mi -I


1 nj-1


Figure 2.4


T%


Ui-1


ui J.












































Figure 2.5







I ni-1
Di1

ri



D I

ti --ni

(a)






Di-_i ni-1


Ui
--i
1____________


Ui


ri I
s i

D i I
n----=ti


-1
1



Di I
t si


Figure 2.6











































Figure 2.7







1 V bl I 1 a bl n1
n2



1 a2
a 12a bc2 b2a


1 b2 n2 C 1 2 b2 n2


a 3 n3

3
n a ________a______


(a) (b)


Figure 2.8












n 2 b0 n c1
1 b11 nI C

b 21 [ . 2 C 2
b21

1 a2_ b 21 n2 C2
C2

1
1 a2 _'__02

b3C3
nn c3


n44 3 31 32




1 W


Figure 2.9









1 bnl

21
21 .._________________ __
b22 :702-------- 2

21 nb212 b22 Cn
1 22 -- 2
b 31

32 c
b33 3
b b34
n3
1 b31 b3 b 43 b34 n3 c3




.. . b43

04
b 46

F e47
S24





Figure 2.10







R l


C. 1


R' R
b+2 Rfni+2
b,=a,.2
''l^ 5!/' 1,2


7=ni


Figure 2.11


C1+21


ci
i+i 1 '' * * -- ^ ni+l

C.1.4 1 ai+l1 ni+l1


ci+2


E .. .. .. J
1---





Ci+l
In i + 1

^JiiiZI

^^^


[__^- ^


2


an1 2
ai2


5i 6f





51

\ ---- 1 "1 1------------I

n

__________3_________________ 02 2
(a) 02 1 n 2

C3 -- 3 f
1 3


Figure 2.12


1 Pi n 1 ____l C 1 P1 nl

P2_________ 02____________________

n2 2 P2
n2, P2 nn2 n2

SP32 ___2___' _P2

S3 __ 1________ 11

n3 n3 P3


(c)

Figure 2.13

















1 p1 1q r,- n1


1 P2q2 2 r2 n2


II

S- (a)
(a)


Figure 2.14


1 Pl q1 r1 nl














CHAPTER 3
PRINCIPAL EMBEDDINGS OF ATRIODIC
PLANE CONTINUE

3.1 Introduction

Several known examples of tree like continue which

admit fixed point free maps are atriodic and nonchainable,

and each proper subcontinuum is an arc ([Bellamy, 1981] ,

[Oversteegen and Rogers, 1980 ] ). In [1980a] Brechner and

Mayer show that if there is an indecomposable nonseparating

plane continuum which admits a fixed point free map, it must

have a Lake-of-Wada channel in every embedding. This is a

consequence of independent results of Bell [1978] and

Sieklucki [1968 ] In this chapter an example is given of a

tree-like, atriodic, nonchainable, indecomposable

nonseparating plane continuum each of whose proper

subcontinua is an arc which has an embedding with a

Lake-of-Wada channel. The construction of the example is

based upon an example of Ingram's [1972] Ingram's example

is a tree-like, atriodic, nonchainable, nonseparating plane

continuum each of whose proper subcontinua is an arc.

However, since it has an embedding with no Lake-of-Wada

channel, it has the fixed point property ([Brechner and

Mayer, 1980a], Theorem 4.1). Both Ingram's example and

ours are proved nonchainable by showing they have a positive

span.






54

We construct our example as an inverse limit of X's, so

will refer to it as the X-odic continuum. The resulting

continuum, X, is homemorphic to a continuum defined as the

intersection of a defining sequence (in the sense of Ingram

and Cook [1967 ]) of tree-covers, and we make use of both

constructions in our proofs. We were informed by C.

Hagopian that he has a similar example as an inverse limit

of X's, but with exactly two Lake-of-Wada channels.

In Section 3.2 we construct the example, X, and prove

that it is atriodic and nonchainable. The main theorems of

this section are 3.2.4 showing that X is atriodic, and 3.2.7

showing that X is nonchainable. The most complex theorem is

3.2.6 showing that X has properties sufficient to guarantee

positive span.

In Section 3.3 we show that X has an embedding in the

plane with a Lake-of-Wada channel. We also raise some

questions about X and the fixed point problem for

nonseparating plane continue.

All spaces are metric and distance functions are as

usual for spaces and their products. All functions (maps)

are continuous.

3.2 The X-odic Continuum.

We shall define the X-odic continuum X in two ways. It

is evident that the continue so defined are homeomorphic.
2
The second definition will be such that X C E and so

provides an embedding of X in the plane.






55

3.2.1 Inverse Limit Definition of X.

Let X1 be the union of intervals [-1,1] on the

coordinate axes in the xy-plane. For convenience we

designate (0,1) as A, (-1,0) as B, (1,0) as C, (0,-i) as D,

and (0,0) as 0. X1 is then the identification of four

intervals, [OA ] [OB ] [OC ] [OD ] at a single point 0.

By P mean the point (0,P) and similarly for B, C, and D.
q q
Thus [0C)] denotes the interval on the x-axis from (0,0) to
q
(. ,0).
q
Let f : X X1 be a map carrying


A
A onto OB, OR (order-reversing and proportionally)



AA
onto OA, OP (order-preserving and proportionally)


AA
3- 2 onto OA, OR


A 2A
2 2A onto OC, OP


2A 5A
-2 -A onto OC, OR
3 6


5A
SA onto OD, OP


B
0O onto OB, OR
3


BB A
2 onto 07, OP


B 2B A
-3 onto OA, OR








B onto OC, OP


0 C onto OB, OR



C C onto OD, OP



D onto OB, OR



D D onto OC, OP



A schematic diagram of f is given in Figure 3.1.



For each i, let X. = X and fi = f. We define X by:

X = lim [ X.,f.3



Let f : X X be defined by
n f
f1 = f ff 2 fn


3.2.2 Defining Sequence for X.

For each n, let Tn be a collection of open disks in the

plane, E such that

(1) T n+ strongly refines Tn; that is, for each L in

T n+ there is some M C T such that L C M.

(2) The mesh of T is less than 1/2n.
n
(3) T is a coherent collection of four subchains with

exactly one junction link designated as follows:






57

(a) The junction link is 0 .

(b) The four endlinks are An, Bn, Cnt and D .

(c) The four subchains are 0 A 0 B n, OC and

0D.
nn
(4) T n+ follows in T the pattern suggested by

function f, including orientation with respect to the

plane. (That is, if T were "straightened out" by some
n2
orientation-preserving homemorphism of E2, then Tn+l

would sit in T exactly as Figure 3.1 suggests.)
n

Then X = R T.
n=l n
Figure 3.2 illustrates the first three stages in the

construction of X. The third stage is represented by its

nerve. Since Tn+l follows the pattern of f in Tn, we may

conveniently refer to links of T on analogy with points of

X That is, A n+/3 is a link of chain 0 n+A n+ that
n n+l n+l n+l
A
corresponds to point (0,h) of Xn. Note An+ /3 is a link of

T n+ sitting in link A of T Similarly, link B n+/2 of
n+1 n n -n+1
T n+l is a subset of link A /3 of T We could have defined
n+1 n n
the pattern that Tn+i follows in T directly in terms of

such intermediate links and the chains between them. We

require our chains have as few bends as possible. (Too many

bends and we might get pseudo-arcs as subcontinua.) We will

call each T (which is a tree cover of X) an X-od cover of
n ---
X. An open cover of X by subsets of X is derived from the
above by letting T' be the collection of open sets of X such
n-

that

L' is in T' iff L' = L n X for some L is in T
n n






58
2
It is clear from our definition that X C E in the

construction of 3.2.2. We will refer to this embedding of X

in what follows.

3.2.3 Definition of the Span of a Function.

The span, of, of function f:X-Y is the least upper

bound of all numbers for which there is a connected subset

Z. of XxX such that I 1(Z ) = f2(Z ) and d(f(x),f(y)) > efor

all (x,y) in Z The span, oX, of space X is the span of

the identity function on X (Ingram [1972] due to Lelek

[1968 ] ).

3.2.4 Theorem

Let X be the continuum defined by X-od tree covers in

3.2.2. Then X is atriodic.

Proof. We will show every proper subcontinuum of X is

chainable. That X is then atriodic follows from Theorem 3

of Ingram [1968].

Let H be a proper subcontinuum of X. Let [T n be
n n=l

the defining sequence of tree-covers for X, defined in

3.2.2. For each n, let F be that subcollection of T that

covers H minimally. Then H = QiF For some N, for all
n~ln
k > N, Fk # Tk, for otherwise H = M. Since H is a

continuum, each F must be coherent. We consider two cases:
n

If for some subsequence [Fj] of (F n) n F does not
include 0., then H = jiF. is chainable as each F. is a

chain.






59

So assume there is no such subsequence. Then for some

J, for all k > J, Ok is in Fk* Let k be chosen as the

maximum of N,J. Then Ok+3 is in Fk+3. Now 0k+3 C Bk+2 in

Fk+2. But 0k+2 is in F k+2. Since Fk+2 is coherent, chain

Ok+2Bk+2 C Fk+2. Observe that

k+2B k+2 C k++l k+U k+1 Ak+l/3 U k+l Ck+l
Ok+lBk+l C 0kBk U Ok+Ak+l/3 U OkCk,

Ok+ A k+/3 C OkBk U OkAk,
Ok+lCk+l C OkBk U OkDk
"k+lk+l kk "kk
Hence, 0 k+2Bk+2 meets every element of Tk. Since

k+2B k+2 C Fk+2 and Fk contains Fk+2' Fk = Tk. But k > N,
so Fk # Tk,a contradiction. Therefore, H is chainable, and

by Ingram's theorem noted above, atriodic. QED

Figure 3.3 shows how 03B3 sits in T.

To show that X is nonchainable we will use Theorem 4 of

Ingram [1972].

3.2.5 Theorem.

If X = lim [ Xi,fi] with each X. compact, and for E> 0,
1
of, >E, for each n, then oX > 0.

Lelek [1968] observes that oX > 0 implies X is

nonchainable.

To apply 3.2.5 we require the following as a lemma.

The proof is an application of the method of proof of

Theorem 2 in Ingram U972].

3.2.6 Theorem.

There exists a sequence Z1, Z2 . of subcontinua of

X1xX1 such that for each n, 1(Zn) = 52(Zn) = XI' fxf(Zn+l)
-l 1
= Zn Z = Zn, and if (p,q) is in Z1, then d(p,q) > -,' for

all n.





60

Proof. Our proof is by induction on n. Z is the

union of the following twenty subcontinua of X1xX I:


m1=([OB]

m3=([OB]
m5=([OC]

m7=([OC]

mg=([ OD]

mll=( [OD]
m13= ( [OC]

ml5=([OD]

m17=([OD]

m 19=([o]


x WA]) U (B] x [OA] ) m2=([OA] x

x [0])U (CB] x [o]) im4=([O] x [B
x [A) U (tC3 x [OA] ) m6=([OA] x

x [3)) U( 3C] x [0o]) m8=([o-] x [C
x WA]) U ((D] x [OA] ) m10=([OA]
x [A3)U((D] x [0) m)12= ( O] x

x (B) U (CC) x [OB] ) m14=([OB]

x (B)) U ((D] x [OB] ) m16=([OB]
x (C3) U ((D] x [OC] ) m18=([OC]
x [A) U (0 x [A] ) m=([5A
x WA) U (L[03 x [5AI m20 [-A])


Each of in. (1 < i <
k (1 j 10) below as
kj (1 <_ j _< 10) below as


[B]) U ([A] x [OB))

I) U ()] x [OB] )
(C]) U ((A) x [OC])


x (D])U(LA] x [OD] )

LD])U(t] x [OD] )
x (C3)U(tB] x [OC]

x (D3)U(LB] x [OD]

x (D])U((C] x [OD]

x (O]XJU(CA] x [o3])


20) is a continuum. We designate

unions of certain m. containing a
1


common point; hence each k. is a contain
3
illustrate ZI.


mI1 U m5 U m9 U mn19
mi2 U m6 U m10 Um20

mi2 U m4 U m13U m115

m U m3 U m 14U m 16
m6 U m8 U ml4U m17

m5 U m7 U m 83U m18

ml 0U m2U m16OU m18
m 9 U m11U m15U mi17


(O,A)
(A, 0)
(O,B)

(B,O)
(O,C)

(CO)
(0,D)
(D,0)


uum. Figures 5 and 6




mI n mi5 f m9n m19

m 2 n m 6 n mlo n m20
m 2 n m4 n m13 n m15
mI fl m3 fl m14n m16

m6 n m8 n m14n m17
inm5 n m7 n m13 n m18

min0 m12 n Am6 mi8
m9 in ml n m15nml17






61

k = m3 U m7 U min (A,) is in m nm n ml
9 3 13 3 1
k1o0= m4 U m U inm12 (,o) is in m4 f n8 m12
Observe that m1 C k 1n k4, m16C k4 n k7,

m10 C k7nki, m6 C k2 n k5, m17 C k5 n kg, m 5 C k8 n k ,
8
m13 C k3nk6. Hence, U k. is a continuum and further (k9U
13 3 j = jl D 9
8 20
k) CU k = Um.. Hence Z = Ui m. is a continuum. We
10 j= j i= iimi
further observe that

For each odd i, (1 < i < 19), mi = mi+1 so 1 = zI

r 1(inm1Unlmlnml3U mi17) = T2(mlU inm10Um13OU m17) = XI,

(B,C) is in m 14, (C,B)is in m13, (B,A) is in mi, (A,B) is

in m2, (B,D) is in m16, (D,B) is in m 15
1
If (p,q) is in ZI, then d(p,q) > 1

We adopt the convention that < t,u > denotes a continuum M

such that 7I1(M) = t and 7T2(M) = u and < t,u >-1 = .

Note that Z1 is the union of 20 such continue in the same

order as in the induction hypothesis below. If < t,u > is a

subcontinuum of Z and v,w are subarcs of X1 such that fiv

maps onto t and fl maps onto u, both homemorphically, then

L = (f-1 Iv x f-11w )(< t, u >) is a continuum such that

TI (L) = v and 72(L) = w.

We denote this continuum as L(, v,w), or more briefly

as L, and call it the lifting of < t,u > with respect to v

and w as defined in Ingram [1971]).

Induction hypothesis. Z is a continuum of X XX1 such

that










(a) r1(Z ) = r2(Zn) = X.

(b) Z is the union of twenty continue denoted by
n
< OB,OA > < OA,OB > < OB,A > < 0O,OB >


< OC,OA > < OA,OC > < OC, A > < O,OC >



< OD,OA > < OA,OD > < OD,OA > < AOD >



< OC,OB > < OB,OC > < OD,OB > < OB,OD >


A 5A 5A A
< OD,OC > < OC,OD > < OA,-A > < -AO >

where < t,u > is a continuum M such that 7I (M) = t and r2(M)

= u.

c) < t,u >-I =< u,t >

d) There are five points in Xl, Z as follows:


5A5A A
x is in --A, with (xl,0) in nnn< A,O>



and (O,xI) in Xnn


2B A
x2 is in 3 B, with (x2,O)in fnn



and (O,x2) in flnfln

CA

x is in -CC, with (x3,O0) in nnln



and (O,x3) in nnn< A,OC>









X4 is in D, with (x4,0) in ffn


and (O,X4) in nnn
3,


x is in with (x,O) in A


and (O,x5) in nn


e) There are three points in X1, Zn as follows:
z is in OC with (B,z ) in and (z1,B) in
z2 is in OA with (B,z2) in
z3 is in OD with (B,z3) in
Base Case. Observe that Z1 meets all of the above
conditions.
Induction. Our induction will be to construct Zn+l by
lifting Zn so Zn+l satisfies (a)-(e) and fxf(z n+) = Z .
n nin+l n
By a. (1 < i < 20) we will denote the twenty continue whose
union is Zn+l* Furthermore, < t,u >' will denote a
continuum which "corresponds" to < t,u > of Z This proof
closely follows Ingram's method in [1972 ]1, as does the
inductive hypothesis above.
Continue a1, a2, a3, and a4.
= a = ' = LI(, -2 B,OA ) UL21 (,- B,A )U


1 A 5A B 2B 11AA 1 A 5A. BB 11AA,
L3 (, -3 363 )UL4 (,",36 3)U


L5(,Ol,) U L (,-2,L)U




64
S1 BOB, 00, B A2A) U LO (2OBA 00 -LA -5A) U

1 B 5A
L9 (,OO, %)

a2 = ' = a1


That a is a continuum follows from


(x3,0) is in n,

so (f-l x3)2B A is in L 1 L
so ',B(3 1 fl2

(O,xI) is in n<0fc ,-6A>,


so (2B, f-1-- (xl)) is in L2nL1


There is a y in 5A, with (A, y) in ,


so (--,f -111A- (y)) is in L31fLl
A 5A
(O,xI) is in <0,-5AA> n,

B h-,IA A L
so (B, j-1-1 A (x1)) is in L L


There is a y in OB, with (y,A) in ,

Sl n1 1
so (f-I ( (y),-) is in L5flL





65
(x2,O) is in n,


so (f1 IO3-(x.2,) is in L6nLl


There is a y in OB with (y,C) in ,

-1B 2A 1 N L1
so (f- I1o (y),--2) is in L71 1l


(x2,O) is in n,


so (f- oI- (x2), 5 ) is in L nLl
So a2 is also a continuum. Also


Tr1(a1) = OB & 72(al) = OC & fxf (aI) C Zn


1(a2) = OC & 7i2(a2) = OB & fxf (a2) C Zn

5
a = 3 where L3 = L1 for 1 < i < 5.
a3 = P OO'=1 iLwhr i i -


a = = a31


The proof that a1(a2) is a continuum contains the proof
that a3(a4) is a continuum. Further
7l(a3) = OB & n2(a3) = & fxf (a3) C Zn
(a4) = & 2(a4) = OB & fxf (a4) C Zn





66
Continue a5, a6, a7, and a8.
5 C A 5(,C AA U
a5= ' = L1(,-=CO) U L ,)

5 C AA 5 C AA
L3(,O-,j) U L4(,0,-|)U

5 C A 2A 5 C 2A5A
L5(,OW,| T-) U L6(,O,-- -5)U

5 C 5A
L7(, O,-:A)

-1
a6 = ' = a5
That a5 is a continuum follows from


(x4,0) is in n,


1 C A5 5
so (f l C x() ,-) is in L5nflL
2o(- (4) 6 L1 L2


(O,x1) is in n,


so (j, f-lli3(x)) is in L 2 L3
2 63 12 3

There is a y in OB with (y,A) in ,
-1C A isi 5
so (f-ioC(y),A) is in L3 nL4


(x2,O0) is in n,


so (f- 1O-(x2) -) is in L 4 L5
7 '4 5





67


There is a y in OB with (y,C) in ,


so (f-1loC(y),2) is in L5
2 is iL5 L6


(x2,O) is in n,


so (f-iCO-(x2),") is in L 5AL


So a6 is also a continuum. Also


TT 1(a5) = OC & 7T 2(a5) = OA & fxf (a5) C Zn

T 1 (a6) = OA & iT 2(a6) = OC & fxf (a6) C Zn


a = ' = UiL7 where L7 = L5 for
7 < = ->il
a = ' = a71

That a7(a8) is a continuum follows from

a5(a6). Further



I1(a8)= OC & IT2(a) =O & fxf


Tr1(a.) = 2& T2(a.) = OC & fxf


1 < i < 3.


the proof for




(a7) C Zn


(a8) C Zn





68
Continue a9, a10, all and a12.

a = ' = L (, DOA) U L9 ( ,-D-)U
9 1
3 2, 9 2 AA
~( , 0j,25 U)
9 ^D A2A. QnA
L5(,Oa,A-2) U L(,O9-, -)U



L9 (^,D 5A.


a = ' = a-
That a9 is a continuum follows from


(x3,0) is in n,


so (f-liD(x3),-) in LInL9


(O,x ) is in n,

s Df 1AA 9n
so (, f --(x)J ) in L2L3


There is a y in OB with (y,A) in ,


so (f-1 1o-(y),2 ) in L9 L9


(x2,0) is in n,


11 D A 9 9
so (f 2(x2),2) in L4 5





69

There is a y in OB with (y,C) in ,


,-l iD, 2A. 9 9
so (f 1lO-(y) ,7) is in L.nLg



(x2,O) is in n,

-1 Dflo 2 5A 9 9
so (f 2O-x) A,-5) is in L6L
So a10 is also a continuum. Also


7T (a9) = OD & T 2(a9) = OA & fxf (a9) C Zn


l (al) = OA &Tr 2(a10) = OD & fxf (a10) C Zn

3
a = a11 = `

a12 = <0 ,OD>' = a-
That a11 (a12) is a continuum follows from proof for a9(a10).
Also

l(all) = OD & 72(al) = O & fxf (a1) C Zn


Ta A (a) OD & fxf (a ) C Z
1Tl(a12) = O & 2(a12) = OD & fxf (a12) C Zn







Continue a13 and a14

13 C B 13 A C BBu
a 3 ' =L1 (,jcO)U L 2( ,JC, ) U


13 A C BB 13 A COB, B>, 2B)U
L 3 (,O-,-2) U L 4


13 C2B
L 5 OBO>O=,-)


-1
a14 = ' = a13
That a13 is a continuum follows from


(x4,O) is in A,
,,lC,, .B, LI 13n 3
so (f -lC(x),) is in L 3nL 3
so4 3 1 2


(O,x5) is in n,
C -f 1BB 13 13
so ( 32f l-jx5)) is in L2 L 3


There is a y in OB with (y,-) in 11 1()B 13 13
so (f-lo(y) ',) is in L3 n L43


(x2,O) is in n,
03>
so (f-l C (x2) 2B- 13 13
= ,-x ) 2 L4 5 L


So a14 is also a continuum. Also


1l(al) = OC & T2(a13) = OB & fxf (a13) C Zn
T 1(a1) = OB & 72(a14) = OC & fxf (a14) C Zn







Continue a15 and a 16.


a ' = LD BIOCOB> 15 A D BB
a15 1 ,D,07 U L 2 '2


L15 OBO>O D BB) U L15 OBA D B 2BU


15 D 2B
L 5(O2-,2i)


-1
a16 = ' = a15


That a15 is a continuum follows from
(x3,O) is in n 35 2 1 5
so (f-1 DD(x3), B) is in L15 n L15


(O,x5) is in ,
D f-i B 15 15
so (D,f f i B(x5)) is in L2 1n L3


There is a y in OB with (y,A) in ,
-1D B 15 15
so (f- lo(y),2-) is in L3 fl L4


(x2,O) is in ,
so (f-1IOE(x ),2--) is in L5 L15
So a16 is also a continuum. Also


T(a15) = OD & I2(a15) = OB & fxf (a15) C Zn


T (a 6) = OB & 7T2(a16) = OD & fxf (a16) C Zn





72

Continue a17 and a18.

a17 = ' = (D U 17 OCOD >,D C )U

1 ,D,O ) U L2 ( 17 1DCO) 2 ~

-17 (,O-,-C)
3 23


a18 = ' = a 17


That a17 is a continuum follows from


(x3,0) is in fl ,

so (f-iID(x3) ,C-) is in L 17 n L 1


(O,x4) is in n ,
,D 1| _, ., . -17r~ -17
so D(-1,f -C(x4)) is in L 17 n L17

So a18 is also a continuum. Also


Tl(a17) = OD & 12(a17) = OC & fxf (a17)


iT (a 1) = OC & TT2(a1g) = OD & fxf (al1)

Continue a19 and a20.


A5A = L19 (,AA A 5A. L19(

5A A -1
a20= < A,0O3> = a19

That a19 is a continuum follows from


(O,x4) is in n ,
(A -i 5A A i i 19 19
so (6',f |-- A(x4)) is in L1 n L2


Cz
n


c z
n


)C >,O-'A






73

So a20 is also a continuum. Also


A 5A
I (a) = 0- & (a 9) = :-A & fxf (ae) C Z
1 19 3 19 6 9

5A A
7T a20) = -6 A & 72 (a = 0i & fxf (a) C Zn
1 20 6 2 (20) 3 (20) n



2 0
We now show Zn = iai is a continuum satisfying (a)
n+l I=
-(e) of the induction hypothesis and such that fxf (Zn+) =

Z Designate the following five points:


x{= 5f-l A = fl f-1 2B(


x = f1 C(z3) & x= f -D(Zl)


f 1 AA
x = f --(z2)



These points will be shown below to satisfy (d) and will be

used to show b. (1 < i < 10), defined below, are each

continue. We explicitly define b. for odd i, and note
1
that for even i, b. = b. .
1 i-11




b = a2 Ua6Ua 10Ua where L implies (x ,0) is in a2, L7


S^10
implies (x ,0) is in a L implies (x{,0) is in ae0,
implies 7XO is in6,L


20
& L 2 implies (x{,0) is in a20.





74

b, = a U a- a Ua where L1 implies (x',0) is in al, L 14
b3 =a1U a14U a16Ua3, where 51


implies~ ~1 implies (x',O) is in ael
16216
implies (x,O) is in a14, L imle5x,)i na6


& L3 implies (x,O) is in a3.


b5 = a5Ua 3Ual8Ua7, where L5 implies (x,0) is in a5 ,


13 18
L 13 implies (xpO) is in a13, L 1 implies (x,O) is in


a 7 & implies (x,) is in a7 .
L18, Lo)


b7= a Ua1Ua1 U a where L9 implies (x.,O) is in a9,
1 5 177ill 1


L 15 implies (x,O) is in ae5, L 17 implies (x.,O)


is in a1, & L 1 implies (x.,O) is in a1l.

=4ipis(,Oisia4 8
b9 = a4Ua8Ua12, where L5 implies (x,,0) is in a ,

2
implies (x,O) is in a8, & L3 implies (x,O) is in a12.


So x{, x x3, xj, x satisfy (d). Observe that

a2C bnb4, a13C b4 n b5, a8 C b5 n b8, a12 C b. n b9,
a8C b9 n b6, a14C b6nb3, a1C b3 b2, ba9 C b2 n b,
2 0
and a1 C b n b Hence Z = a is a continuum, and
is the union of twenty continue satisfying (b).
is the union of twenty continue satisfying (b).






75

Note that T1(a1 U a13 U a17 U a10) = 2 (a1 U al3 U a17U

a10) = XI, so (a) is also satisfied by Z n=. Furthermore,

(e) is satisfied since


There is a z' in OC with (B,z') in al4= ' and (z{,B)

in ae3 = '.


There is a z in OA with (B,z2) in a = ' and (z-,B)

in a2 ='


There is a z' in OD in (B,z') in a 6= ' and (zj,B) in

a15 = '


-I
As a. = a11 for odd i, 1 < i < 19, (c) is satisfied by
i l
Zn+1

Finally, we must show fxf (Zn+l) = Zn. Since fxf (a.)

CZ for 1 < i < 20, we need only show Z C fxf (Z n+l). We

observe that fxf (a3 U a13 U al5 U a18 U a20) includes


A A
, ,


, , , ,
Hence fxf (a4 a U 14 a6 U ae7 U ael 9) includes


, , , < OC>,


< > <5A A
<9,OD>, , , , <6AOr>.






76

Thus Zn C fxf (Z n+l). Therefore, fxf (Z n+) = Z .
n n+l n+l n
Our inductive step is thereby completed. Since (p,q)
1 n 1
in Z1 implies d(p,q) > it follows that of > 3, for all n

and the theorem is proved. QED

Figure 3.5 represents Z1 C X XX1 with hatched and x-ed

lines.

Figure 3.7 represents Z2 C XlXX1 with hatched lines

going to hatched lines and x-ed lines going to x-ed lines in

Figure 5 under fxf.

Two parts of Z3 are illustrated at the bottom of Figure

3.7.

3.2.7 Theorem.

X is nonchainable.

Proof. By Theorem 3.2.6,oX satisfies the conditions of

Theorem 3.2.5. Hence we may conclude that OX > 0.

Therefore, X is nonchainable. QED

That X is nonseparating is evident. That X is

indecomposable is a corollary to the proof of Theorem 3.2.3

by applying Ingram and Cook's criterion of indecomposability

[1967] Another proof that X is indecomposable follows from

Theorem 3.3.2 and a theorem of Sieklucki [1968] quoted in

Lemma 2.2 of Brechner and Mayer [1980a]. It is evident from

the construction of X that each proper subcontinuum, being

chainable, is also an arc, since all subchains are

relatively straight in the covers they refine.

We observe that X has two and only two endpoints, e and

f, such that e is the intersection of the tower






77

C . CDC C4 C D3 C C2C D1



and f is the intersection of the tower



C . CC5C D4 C C3 C D2C C1



3.3. Lake-of-Wada Channels.

By Lake-of-Wada channel we mean a simple canal,

defined in Brechner and Mayer [1980a] which definition is

due to Sieklucki [1968] (For the original Lake-of-Wada

construction see Hocking and Young [1961] pp. 143-144.)

3.3.1 Definition.

Let X be a nonseparating continuum and let D be a set

homemorphic to [0,1) in E X, where a:D -* [0,1) is a

given homeomorphism. Then D will be called a simple canal

in X if the following three conditions are satisfied:



(1) D D C Bd X

(2) For each p in D, there is a "bridge" to X; that is

a crosscut to X, which (crosscut) is transverse to D,

and intersects D at exactly one point.

(3) If p. (i.e., a(pi) + 1), then there is a

sequence of bridges (Q }i such that Q Pi D = (pi) and

diam QPi 0.
Pi
If, in addition, condition (4) holds, where (4) is

(4) D D = Bd X,

we call D a simple dense canal (s-d-c) [Sieklucki, 1968].





78

We can show that the embedding of X given in 3.2.2 has

a s-d-c or Lake-of-Wada channel in either of two ways: by
2
directly constructing the requisite ray in E -X, or by

defining a chain of crosscuts Q of E -x that defines

a prime end E of the third kind with I(E)=P(E)=X. (See

Brechner [1978] for basic definition of prime ends.) For

the equivalence of these methods see Brechner and Mayer

[1980a], Theorem 2.9. An embedding for which there exists a

s-d-c D in E -X such that D D = Bd X, is termed principal

in Definition 2.6 [Brechner and Mayer, 1980a] The embedding

of X here given is consequently principal.

The first four stages of the construction of a s-d-c D

are illustrated in Figure 3.4. Figure 3.2 illustrates two

stages of D (the "railroad tracks"). Alternately, let Qi be

a crosscut of E -X such that for all i > 2, the endpoints of

Qi lie in C. and A., while Qi C D. i. Then either the odd

or the even subsequence of (Q 3' defines a prime end E
i i=2
such that I(E)=P(E)=X. For the even subsequence, Qi --e, and

for the odd subsequence, Q. --f, where e and f are the

endpoints of X. We thus have the theorem below.

3.3.2 Theorem.

X can be embedded with a simple dense canal D in E2-X

such that D D = X. That is, the embedding of 3.2.2 is

principal.

3.3.3 Corollary.

X is indecomposable.

Proof. Follows from 3.3.1 and Lemma 2.2 of Brechner

and Mayer [1980a].






79

If every embedding of X into the plane were such that

E -X contained a s-d-c, then X would be a principal

continuum in the sense of Definition 2.6 of Brechner and

Mayer [1980a]. As a principal continuum X would be a

candidate for a nonseparating plane continuum admitting a

fixed point free map. Thus two questions suggest

themselves

3.3.4 Question.

Is X a principal continuum?

3.3.5 Question.

Does X have the fixed point property for continuous

maps?

In [1975] Lelek asks if there is an example of a

nonchainable continuum with span zero. This author, in

efforts to modify Ingram's example to produce a principally

embedded, atriodic, nonchainable continuum considered

several examples for which said author was not able to prove

the span nonzero. On the other hand, the examples are

principally embedded, and appear to be nonchainable, though

atriodic. We show a schematic diagram for the bonding

function for one such example as an inverse limit of T's in

Figure 3.8. (For surjective span see Lelek [1977].)

3.3.6 Question.

Does the continuum of Figure 3.8 have a span greater

than zero? Surjective span greater than zero?

3.3.7 Question.

Is the continuum of Figure 3.8 chainable?
















































D


Figure 3.1

























































Figure 3.2




























Figure 3.3


Figure 3.4























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B -

C. C







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A B C. PB PB






I B
B




a

A C-

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a
'p.'
C



CI





C
0

0




4,


B
C

D ~

at


Figure 3.5


fll

























































Part of X1 x X1 with part of Z1 marked

Figure 3.6




















... i : . . . ___ . .. k ,
A A


A " a~~a O a a- Ce "- 0--- ^ ^ -- '
Sal






A o C .o ,- D-








I l r ....... D ,L '' D- | ,\
I I I i . . .
.. .. ...- --- -- ----- --- :,,-,-----,----- '*' '






0 0 0
B 0.' 0 C. 0

~ ~ .. .. ,..t




o '. *. o
-o .'
A ,. f ^ ^ ^ o ot6


Figure 3.7

















































Figure 3.8














CHAPTER 4
INEQUIVALENT EMBEDDINGS
AND PRIME ENDS

4.1 INTRODUCTION.

In [1980a] Brechner and Mayer show that equivalent

embeddings of a nonseparating plane continuum have the same

prime end structure (Theorem 2.11). Though not explicitly

stated, this fact has been used previously in the

literature. For instance, M. Smith [1980] and W. Lewis

[1980] have independently shown that there are uncountably

many inequivalent embeddings of the pseudo arc into the

plane. This result was achieved by exploiting different

prime end structures (directly in Lewis' case, indirectly,

in terms of differing accessibility of composants by Smith)

to distinguish different embeddings.

In this Chapter we show that the converse of Brechner

and Mayer's theorem is false: there are inequivalent

embeddings of a nonseparating continuum into the plane that

have the same prime end structure, and indeed, that have the

same set of accessible points. The following theorems stand

in partial contrast to the methods of Smith and Lewis:

4.1.1 Theorem.

There exist uncountably many inequivalent embeddings of

the sin 1/x continuum into the plane with the same prime end

structure and the same set of accessible points.

87








4.1.2 Theorem.

There exist uncountably many inequivalent embeddings of

the Knaster U-continuum (bucket handle) into the plane with

the same prime end structure. Moreover, the set of

accessible points in each of these embeddings is exactly the

composant of the U-continuum that contains the endpoint of

the U-continuum.

In Section 4.2, we show how to construct uncountably

many embeddings of the continuum formed by the sin 1/x curve

plus its limit segment. Thereafter, we show these

embeddings have the same prime end structure, but that any

two are inequivalent, thus proving Theorem 4.1.1.

In Section 4.3, we proceed similarly to prove Theorem

4.1.2, showing the Knaster U-continuum also has uncountably

many inequivalent embeddings with the same prime end

structure and the same set of accessible points. We first

prove the theorem for two specific, easily visualizable,

embeddings of the U-continuum.

In Section 4.4, we indicate how Theorem 4.1.2 can be

extended to each of the uncountable class of U-type Knaster

continue identified by W.T. Watkins [1980] .

4.2 The Sin 1/x Continuum.

The standard embedding of the sin 1/x continuum (Figure

4.1) consists of a ray R, the graph of (0,1] under the

function y=sin 1/x in the xy-plane, plus the limit segment

[p,q] the interval [-1,1] on the y-axis. The ray R

consists of a number of loops, where a loop is a segment of

R with exactly one peak and one trough in the standard






89

embedding. For simplicity, we will fix the set of loops as

the segments of R between alternate successive points of

zero amplitude in order from 1 to 0 on the x-axis. The

endpoints of loops limit on (0,0) in the standard embedding,

so on the "midpoint" (or, more precisely, some interior

point) of [p,q] in any other embedding. Endpoint p of limit

segment [p,q] is the limit point of points on R selected

from successive troughs, endpoint q is the limit point of

points selected from successive peaks. We suppose R to be

coordinatized by the function g: [0,) -* R so that the

endpoint of R is g(0) and for x E (0,1] odd positive

integers correspond to t(x,l): sin 1/x = 13, even positive

integers correspond to t(x,-l): sin 1/x = -13, and fractions

with denominator 2 and odd numerator greater than 1

correspond to [(x,0): x=l/(in), for positive integers n].

This embedding, denoted K, and its corrdinatization by g are

illustrated in Figure 4.1.

An embedding e:X E 2 of a continuum X is a

homeomorphism into the plane; however, we shall somewhat

loosely suppress reference to an embedding function and

refer to the image in the plane of a continuum X as the

embedding of X.

An embedding of the sin 1/x continuum can be described

in terms of how the limit segment and each loop

L = [(2n-l)/2, n, (2n+l)/2, n+l, (2n+3)/2 ]
n
for odd positive integers n, is embedded in the plane. We

will continue to designate the image of R, [p,q] Ln, or a

point x in the sin 1/x continuum as R, [p,q] L or x,






90

suppressing reference to any particular embedding function.

A schema will be a set of directions (necessarily infinite,

though countable) for embedding (and, ambiguously, the

embedding of) the loops of ray R with respect to limit

segment [p,q]. A subschema will be directions for embedding

some finite number of loops. Subschemata will be linked in

sequence to form a schema.

We will reference the various subschemata rather

informally. An example will indicate our procedure. We

assume the limit segment [p,q] is fixed, and describe the

embedding of the loops with reference to [p,q] The

embedding M0 in Figure 4.2 is the simplest of the

uncountably many we will show to exist. For this embedding

we require only one subschema So: from (2n-l)/2 bend around

q at n, then toward, but not around p at n+l, and extend to

(2n+3)/2. The schema P0 for embedding M0 is then the

infinite sequence of So's linked so that corresponding parts

of successive subschemata are closer to [p,q] than their

predecessors.

4.2.1 The Prime End Structures of K and MO.

Before proceeding to construct our uncountably many

embeddings of the sin 1/x continuum, we illustrate some of

the concepts involved in prime end theory by applying them

to K and MO. Definitions and further references may be

found in Brechner [19781 .

Prime ends are a way of looking at and classifying the

approaches to the boundary of a simply connected domain with

nondegenerate boundary. The complement in S of a






91

nonseparating nondegenerate plane continuum X, denoted S2-X,

is a simply connected domain. While E -X is not simply

connected, as E2U[,), the one-point compactification of E2,
2 2
is S we can refer to the prime end structure of E -X by

associating it with the prime end structure of S -X, where

the embedding at X misses the point at infinity.

A prime end of E -X is defined by a chain of crosscuts

converging to a point of X, where a crosscut is an open arc

in E -X whose endpoints lie in X. If Q is a crosscut of
2_ 2
E -X, then QUX separates E A sequence of crosscuts

(Q.i.i=l is a chain provided that Q. converges to a point,

that no two crosscuts have a common endpoint, and that Qi

separates Q i-i and Qi+l. So, for example, the prime ends E

and F of E -K in Figure 4.1 are defined by chains of

crosscuts (Q..il and (Q.)i=l respectively, while in

Figure 4.2, [T.] i= defines prime end H of E2

The impression of a prime end E, denoted I(E), is the

intersection of the closures of the domains cut off by the

crosscuts in a chain defining E. For example, in Figure 4.1

it can be noted that I(E)=[p,q] =I(F), while in Figure 4.2,

I (H) =p,q ] .

The set of principal points of a prime end E, denoted

P(E), is the collection of all points in X to which some

chain of crosscuts defining E converges. For example, in

Figure 4.1, P(E)={p), and P(F)=[q], while in Figure 4.2,

P(H)=[p3.

A prime end E is of the first kind if I(E)=P(E), both

degenerate, of the second kind if I(E)#P(E), only P(E)






92

degenerate, of the third kind if I(E)=P(E), both

nondegenerate, and of the fourth kind if I(E)#P(E), both

nondegenerate. It can be shown that P(E)CI(E) in any case,

and that both are continue in X. Thus prime ends E, F, and

H of Figures 4.1 and 4.2 are all of the second kind. Any

other prime end G, of either E -K or E -M0, will be of the

first kind, or trivial. Thus we can say that the prime end

structure of E2-K consists of two prime ends of the second

kind and all other prime ends trivial.

A more precise description of prime end structure is

afforded by the notion of a C-map (see Brechner [1978].) A

C-map p is a homeomorphism of S -X onto Ext B, where Ext B

is the complementary domain of the unit disk B in S which

contains the point at infinity, which (map) satisfies the

conditions:

(1) if Q is a crosscut of S2-X, then O(Q) is a

crosscut of Ext B, and

(2) the endpoints of images of crosscuts are dense in

Bd B, the boundary of B.

If we require, as we may, that 0 take the point at infinity

in S 2-X to the point at infinity in Ext B, then we can

regard the restriction of 1 to S2-t_}]=E2 as a C-map also.

Suppose a chain of crosscuts defines a prime end E of

E -X. Then the images of the crosscuts under a C-map 0 will

converge to a single point e in Bd B. We say e corresponds

to E. In fact, there is a one-to-one correspondence between

the prime ends of E2-X and the points of Bd B. For example,





93

in Figure 4.1, points e and f in Bd B correspond to prime

ends E and F, respectively. In Figure 4.2, h corresponds to

H. No homeomorphism of Bd B in Figure 4.1 to Bd B in Figure

4.2 can carry points corresponding to prime ends of a given

kind onto points corresponding to prime ends of the same

kind. Hence the prime end structures of K and M are not

identical.(See Brechner and Mayer [1980a], Definition 2.10.)

Consequently, K and M0 are inequivalently embedded. (See

Brechner and Mayer [1980a], Theorem 2.11.)

In the general result which follows, each embedding of

the sin 1/x continuum will have the same prime end structure

as M Note that the accessible points of M0 are the ray R

and point p of [p,q] A point of a plane continuum X is

accessible if it can be reached by a half open arc in the

complement whose closure adds exactly that point. Such a

half open arc is called an endcut. Each of our uncountably

many embeddings of the sin 1/x continuum will have the same

set of accessible points as Mo. The significance of this

result is the contrast it provides to the usual procedure

for producing inequivalent embeddings of a plane continuum:

produce embeddings with different points accessible. Such a

procedure is a sufficient, but not a necessary, condition
2
for producing inequivalent embeddings of a continuum in E.

In order for subschemata to be constituents of a schema

for embedding the sin 1/x continuum, the subschemata must be

linkable into a ray that converges to limit segment [p,q ] .

Only certain subschemata are so linkable, and the following

Lemma identifies countably many of them.








4.2.2 Lemma.

There exist a countable infinity of subschemata

linkable pairwise in either order.

Proof. In the following, all bends in loops are either

around q, or toward, but not around p, or away from p. We

will describe the first few subschemata fully, and then

supress reference to q-bends and designate p-bends as either

in or out with a superscript for the number of bends.

Let n be a positive odd integer. The subschemata are:

So: from (2n-l)/2 bend around q at n, then toward, but

not around p at n+l, and extend to (2n+3)/2. (1

loop: in .)

S1: from (2n-l)/2 bend around q at n, then away from p

at n+1, extending to (2n+3)/2; then bend around q

at n+2, away from p at n+3, to (2n+7)/2; then bend

around q at n+4, toward, but not around p at n+5,

to (2n+ll)/2 which will lie between [p,q] and
2 1
previous loops. (3 loops: out in .)
4 1
S$:5 loops: out ,in.






Sk: 2k+l loops: out2k, in1
Sk : 2k+l loops: out ,in


Subschemata S1 and S2 are illustrated in Figure 4.3.