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Relations on spaces

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Title:
Relations on spaces
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Lin, Shwu-Yeng Tzeng, 1934-
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iii, 51 leaves : illus. ; 28 cm.

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Distance functions ( jstor )
Hausdorff spaces ( jstor )
Homomorphisms ( jstor )
Integers ( jstor )
Mathematical theorems ( jstor )
Mathematics ( jstor )
Semigroups ( jstor )
Topological spaces ( jstor )
Topological theorems ( jstor )
Topology ( jstor )
Dissertations, Academic -- Mathematics -- UF
Generalized spaces ( lcsh )
Mathematics thesis Ph. D
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Thesis:
Thesis - University of Florida.
Bibliography:
Bibliography: leaves 48-50.
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Manuscript copy.
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Vita.

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Full Text


RELATIONS ON SPACES
By
SHWU-YENG TZENG LIN
A 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
August, 1965


ACKNOWLEDGMENTS
The author wishes to acknowledge her indebtedness
and express her sincere gratitude to Dr. A. D. Wallace,
Professor of the Department of Mathematics and Chairman
of her Supervisory Committee, for his guidance and sugges
tions throughout the preparation of this dissertation.
She wishes to express her deep appreciation to Professor
A. R. Bednarek, who read the entire manuscript in detail
and made numerous corrections and improvements. Thanks
are due also to all members of the Supervisory Committee
for their encouragements.
The author is grateful to Dr. D. C. Rose for
correcting the language. She is also grateful to her
husband Dr. Y.-F. Lin, who was always patient in discus
sing mathematics with her and who rendered many valuable
suggestions. And she extends her thanks to Mrs. K. P.
Grady for her excellent work in typing the manuscript.
ii


TABLE OP CONTENTS
Page
ACKNOWLEDGMENTS ii
Chapter
I.PRELIMINARIES 1
II.A CHARACTERIZATION OF THE OUTPOINT
ORDER ON A TREE 11
III.FIXED POINTS AND MINIMAL ELEMENTS 25
IV.FIXED POINT PROPERTIES AND INVERSE
LIMIT SPACES 38
BIBLIOGRAPHY 48
BIOGRAPHICAL SKETCH 51
iii


CHAPTER I
PRELIMINARIES
A relation R on a topological space X is_ a_
subset of X x X. If A is a subset of X and if R is
a relation on X, let
AR = p2 ((AXX)^R)
and
RA = Pl ((X x A) r\ R)
where p^, p2 are respectively the first and the second
projections. For x in X we shall write simply xR for
{ x } R.
Let
R
(-D
= { (y, x) | (x, y) e R ) ,
and
AR^ { x | xR C A } ,
R["1]A = { y | Ry C A } .
In all that follows, X will always denote a Hausdorff
space, and R a relation on X. The closure of a set A
will be denoted by A* and the interior of A by A .
Definition 1.1. A relation R on X is
said to be
lower semi-continuous (abbreviated l.s.c.)
upper semi-continuous (abbreviated u.s.c.)
1


2
at x in X if and only if x in
(vr(-1)) >
l for all V = Vo C X. R is said to be
(W 0 J
.l.s.c. ,
on X if and only if R is
, l.S.C. \
at
u.s.c. /
* U.S.C. i
x for all x in X. R is said to be continuous if and only
if R is both l.s.c. and u.s.c.
Definition 1.2. A relation R on X is
, upper
semiclosed
(u. s .cl.) -j
V if and only if
n
\ lower
semiclosed
(l.s.cl.)J
^ Rx -J
is closed
for any x
in X.
Definition
1.3* A relation R on X
is said to
be a partial order1 if and only if the following conditions
are satisfied:
(a) (reflexivity): (x, x) £ R for all x in X,
(b) (antisymmetry): (x, y) 6 R and (y, x) £ R
imply x = y,
(c) (transitivity): (x, y) 6 R and (y, z) £ R
imply (x, z) £ R.
VR
VR
(-D
[-1]
- implies x in
1. A relation R is said to be a quasi-order if
(a) and (c) are satisfied.


3
We here state some theorems to which we will refer
throughout this work. The proofs of these theorems may be
found in [19].
Theorem 1.1. If R is closed in X x x and if
A is compact, then both AR and RA are closed; moreover,
if V is an open set then R^-^ V and VR^-1^ are open.
Theorem 1.2. The following statements are equi
valent :
(a) R is l.s.c. on X.
(b) RA £ (RA) for all A C X.
(c) A* R C (AR) for all AC X.
(d) RV is open for all open VC X.
(e) p^ | R is open, where p^ | R is the
restriction of p^ to R.
Theorem 1.3- The following statements are equi
valent :
(a) R is u.s.c.
(b) VR^"1^ is open for all V C X.
(c) AR^-1) is closed for all closed A C X.
(d) A R^"1^ C (AR[_1]) for any A C X.
(e) (AR^_1^)^C A* r(-1) for any A C X.
Theorem 1.4. If X is a compact Hausdorff space,


4
then R = R* if and only if
(a) R is u.s.c. on X, and
(b) R is u.s.cl. on X.
Definition 1.4. A subset C of X is an R-chain
if and only if C x c C R U R^-1^.
As a consequence of the well-known Hausdorff
maximality principle, there is a maximal R-chain for any
relation R on any space X. A useful result of
Wallace [14] is the following.
Theorem 1.5- If R is both u.s.cl. and l.s.cl.
quasi-order, then every maximal R-chain is closed.
Definition 1.5- An element a in X is R-minimal
(R-maximal), whenever (x, a) e R ((a, x) e R) implies
(a, x) e R ((x, a) e r).
The following fundamental theorem first proved in
[14] will be used repeatedly.
Theorem 1.6. If R is a u.s.cl. (l.s.cl.)
relation on a compact space X, and if A is a non-void
closed subset of X, then A contains an R-maximal
(R-minimal) element a of A.


The Alexander-Kolmogoroff cohomology groups will
he used as developed in [21]. In what follows the coef
ficient group is fixed and therefore will not be mentioned.
We record here some useful theorems from [21].
Theorem 1.7. If B C A C X, then
H(X,A) -X H(X,B) H(A,B) >
H1(X, A) -^H1(X,B) ^>H1(A,B) *
6
^
Hn(X, A) J >
is an exact sequence, where i* and j* are the induced
homomorphisms of the inclusion maps i and j from A to
X and X to X, respectively, and 6 is the coboundary
operator for the triple (X, A, B).
Theorem 1.8. For the space X and any connected
set A of X the homomorphism i* of
H(X) -X H( A) X H1(X, A)
is an epimorphism and hence 6 is 0.
For simplicity we denote by f:(X, A) - (Y, B)
that f is a_ function from X to_ Y and A is. £ subset
of X such that f(A) C B C Y. Furthermore, if
i:(X, A) -> (X, B) is_ an inclusion map, we will write


6
i:(X, A) C (X, B).
Theorem 1.9.
continuous, and if u:
v:
w:
If f: (X, A, B) -> (X' A'
(X, A) (X' A') ,
(X, B) -> (X', B' ) ,
(A, B) -> (A', B')
B' ) is
are defined by
u(x) = v(x) = w(x) = f(x),
then the ladder
Hp(X, A) HP(X, B)
u1
-^->HP(A, B)
I
w
-^->HP(X, A)^HP(X, B)^HP(A, B) 3--
is analytic, that is, each rectangle of the ladder is
analytic.
We state here a special case of The Mayer-Vietoris
Sequence which will be sufficient in what follows for our
purpose.
Theorem 1.10. If X is a compact Hausdorff space,
and if X = X-^ Xg where X^ and Xg are closed subsets
of X, then there exists an exact sequence
A Hq_1(X1 r\ X2) -A> Hq(x) -i: Hq(X1) X Hq(Xg)
Hq(xx r\ x2) -A, ...


7
where (j* i* 6 ) and (j* i* ) are the homo-
v a a cr a or
morphisms in the exact sequences for the triples (X, X )
and
(Xa, Xx X2
, a
)
respectively for
k:
(X2, X1 kh X2)
c
(x,
x1) ; J* = x
I*
_ i* 1* .
- 1 1 1 2
and
A
i* v* 1 T
Jl k 2
As a consequence of the foregoing theorem we have
the following
Theorem 1.11. With the hypotheses of Theorem
1.10 and if X-^ r\ Xg is connected, the homomorphism A in
h(x1 r\ x2) -A* h1(x) -A_> h1(x1) X h1(x2)
is 0.
Proof. Observe the following ladder
H(x) 2+ H1(X, Xx) ^ H1(X) >
u* k*
'r j j* -7- *
h(x1 ^ x2) % h1(x2, x1 r\ x2) % hx(x2) ^
where u: Xx a X2 C X and where (j*, i*, 6]L) / (J*, Tg, ^)
and k are as in Theorem 1.10. It follows from Theorem 1.8
that if h is in H(X^ f\ Xg) then there is an h' in
H(X) such that u*(h') = h. Thus by Theorem 1.9., we have


8
k* -1 2(h) = k* k* 61(h/ )

which contains 1(h/). And A = j k* 6^ being
a well-defined homomorphism, we see that for any a and 8
in k -1 6*2(h) j*(a) = j*(p). Therefore,
A(h) = J* k* _1 2(h) = j^61(h/) = 0.
The following notation is convenient. _If P
is a_ subset of Q and if h e Hp(Q) then h | P denotes
the image of h under the natural homomorphism induced by
the inclusion map of P into Q.
Theorem 1.12. (Reduction Theorem). If
compact Hausdorff space, if A is closed, and if
such that h | A = 0, then there is an open set U
A such that h | U =0.
Definition 1.6. (X, A) is a compact pair if and
only if X is a compact Hausdorff space and A is a closed
subset of X.
X is a
h e HP(X)
containing
Theorem 1.13 (Map Excision Theorem). If (X, A)
and (Y, B) are compact pairs, and if
f: (X, A) -> (Y, B)
is a closed map such that f takes X A topologically
onto Y B, then


9
f* : HP(Y, B) = HP(X, A).
\ *
r *
Theorem 1.14. (Homotopy Lemma). If (X, A) is
a compact pair and if T is a connected space, and if for
each t in T
At : (X, A) -> (X x t, A x t)
is defined by A^(x, t) = (x, t),
then A = A"'*' for r and s any elements of T.
I? s
Definition 1.7 A space X is unicoherent if
and only if X is connected and X = A B with A and
B closed and connected implies A rs B is connected. X
2
is hereditarily unicoherent if every subcontinuum of X
is unicoherent.
Theorem 1.15- If X is a continuum and if
H1(X) = 0 for G £ 0, then X is unicoherent.
Definition 1.8. If X is a space, if A C X
and if h is a non-zero member of HP(A), then a closed
set P C A is a floor for h if and only if h | F M
while h | F' = 0 for any closed proper subset F' of F.
space.
2. A continuum is a compact connected Hausdorff


Theorem 1.16. (Floor Theorem). If (X, A)
is a compact pair and If h is a non-zero member of
HP(A), then h has a floor. Moreover, every floor is
connected.
Theorem 1.17 If (X, A) is a compact pair, if
R = A x A U A where A= { (x, x) [ x e X } then
HP(X, A) HP(X/R) for all p> 1.


CHAPTER II
A CHARACTERIZATION OF THE OUTPOINT ORDER
ON A TREE
A tree is a continuum such that every two
distinct points are separated by the omission of a third
point. Let X be a tree and let z be an arbitrary,
but fixed element of X. Let Q(z) be the set of all
such pairs (a, b) in X X such that at least one of
the following three conditions is satisfied :
(i) a z,
(ii) a = b, or
(iii) a separates z and b in X.
It turns out that Q(z) is a continuous partial order on
X, and with respect to this partial order z is the unique
minimal element. We shall refer to Q(z), for any z in
X, as a cutpoint-order [l] on the tree X. The purpose
of this chapter is to give a characterization of the cut-
point order on a tree (Theorem 2.4). We also obtain a
new characterization of a tree from Relation-theoretic and
Cohomological view-points.
11


12
Several characterizations of a tree have been
given [2], [5], [23], and [24]. Perhaps the most useful
of these characterizations is
Lemma 2.1. [2], [5]- A continuum X is a tree
if and only if it is locally connected and hereditarily
unicoherent.
An excellent proof of this lemma may be found
in Ward [24].
Definition 2.1. A space X is said to be
semi-locally-connected (abbreviated s.l.c.) at a point x
of X provided for any open set U in X containing x
there exists an open set V containing x such that
V C U and that X V has only a finite number of
components. If X is s.l.c. at each of its points, it is
said to be s.l.c.
In 1953> A. D. Wallace [15] proved that one-
cod imensional"*" compact connected and locally connected
topological semigroup with unit and zero is a tree.
L. W. Anderson and L. E. Ward, Jr. in 1961 [l] modified
Wallace's result by eliminating the necessity of hypothe
sizing a unit. More precisely, they proved that if
1. For the definition and properties of Codimension,
see Wallace [21], or Cohen [6].


13
X is a compact connected, locally connected, one-
codimensional topological semilattice, then X. is a tree.
Wallace [17] improved this result by weakening the local
connectedness of X to semilocal connectedness of X.
These elegant results on Topological Algebra, motivated
the following Lemma which bears a Relation-theoretic
analogy.
Lemma 2.2. If R is a relation on a compact
Hausdorff space X with RX a one-codimensional semi-
locally connected subspace, such that
/ \ *
(i)the relation R is closed, i.e., R = R ,
(ii)H^Rx) = 0 for every x in X,
(iii)the collection j Rx | x e X^ has the finite
intersection property (abbreviated f.i.p.), and
(iv)Ra r\ Rb is connected for each pair a, b
in X,
then RS is a tree for every closed subset S
of X.
The proof of Lemma 2.2. depends on the following:
Lemma 2.3- If A, M and B are disjoint non-void
closed subsets of a normal space X, and if A is either
compact or consisting of finitely many components such that
(i) M does not separate A and B in X, and (ii) for


14
any open set U containing M there is an open subset V
of U containing M such that X V has only a finite
number of components, then there exists a closed and
connected subset N of X such that N <, X M and N
meets both A and B.
This lemma was first proved by G. T. Whyburn [26]
for the particular case in which X was assumed to be a
metric continuum and card A = card M = card B = 1. The
non-metric case was implicit in a paper by Wallace [17]
but without proof. We postpone the proof of Lemma 2.3* to
the end of this chapter.
Proof of Lemma 2.2. It follows from (i) and
Theorem 1.1. that RX is closed and from (iii) and (iv)
that
RX = (Rx |x e X)
is connected and thus RX is a continuum. Similarly RS
is a continuum.
Since every subcontinuum of a tree is itself a
tree and since RS is a subcontinuum of RX, it is
sufficient to show that RX is a tree.
We first show H^(RX) =0. If there were a
non-zero h e H1(RX), then there would be a maximal (non
void) tower J of closed subsets A of X such that
h i RA ? 0. Let Aq= r\ { A | A 7 j
Then


15
h ¡ RAq 0, for if h | RAq = 0, then by the Reduction
Theorem (Chapter 1, Theorem 1.12.) there would,', be an open
V 1 RA such that h I V* = 0. It would then follow
* o 1
from Theorem 1.1. that R^-1^ V would be an open set
containing AQ. If R^"^V is designated by U then
R U C V so that there is an A in J with A < U and
RA C RU C V*; therefore h [ RA = 0, a contradiction.
Case 1. Card Aq = 1, i.e., Aq = { x } By
(ii) H^(RA ) = 0, a contradiction.
Case 2. Card Aq > 1. Write Aq = A^ U A0
where both A^ and A2 are proper closed subsets of A .
We consider the following part of the Mayer-Vietoris exact
sequence (Chapter I, Theorem 1.10.),
h(ra1 n ra2) A, h1(rao) _£*, h1(ra1) x h1(ra2).
Since by (iii) and (iv)
RA1 r\ RAg = IJ | RA a Rb | (a, b) £ A^ x Ag | is
connected, then A = 0 (Theorem 1.11.), and
h RA e Ker J = ImA = 0,
o
a contradiction.
RX is a continuum and H^RX) = 0 imply that
RX is unicoherent (Theorem 1.15*) RX being of codimension
one and H1(RX) = 0 imply that H^K) = 0 for every closed
subset K of RX [21], and thus every subcontinuum of RX
is unicoherent.


16
We now prove that every two points of RX are
separated in RX by a third point. Suppose there were
two points a and b such that no point separated a
and b in RX. Then by Lemma 2.3-, for any p different
from both a and b, there would be a continuum P which
would be irreducible from a to b and which would not
contain p. If q were an element of P distinct from
a and b there would also be a continuum Q, irreducible
from a to b and which would not contain q. But then
P Q would be a subcontinuum of RX which would not be
unicoherent, since P n Q by our selection of P and Q
is obviously not connected. This contradiction completes
the proof.
Theorem 2.4. If X is a compact Hausdorff
space, and if P is a relation on X, then the following
conditions
(i)X is of 1-codimension and s.l.c.,
(ii)P is a closed partial order,
(iii)P is left monotone, i.e., Px is connected,
and H^(Px) = 0 for every x in X,
(iv){Px | x e X) has the f.i.p., and
(v)P is right monotone, i.e., xP is
connected for every x in X,
are necessary and sufficient conditions that X be a


17
p
tree, and that P be a cutpolnt-order.
r
Proof: We first prove the sufficiency.
Conditions (ii), (iv) and the first half of (iii) imply
that
Pa Pb = U { Px I x e Pa r\ Pb }
is connected, and thus Lemma 2.2. yields that X is a
tree.
Since X is compact and { Px I x £ X} has the
f.i.p., then
{ Px | x e x } £
Indeed, it is a single point, the unique P-minimal element
of X. Let us denote by 0 the set
{ Px I x e X } .
We prove that P = Q(0) If (a, b) £ Q(0) such that
a = 0 or a = b, then clearly (a, b) must be also in P.
If a separates 0 and b in X, then since Pb is a
continuum containing 0 and b, it must contain a, and
we again conclude that (a, b) is in P. Thus Q(0) C P.
Conversely, if (a, b) is in P, then since a is in
aP r\ Pb, and since both aP and Pb are continua, then
aP ^ Pb is a subcontinuum of the tree X, and therefore
2. Condition (v) is not necessary for X to be
a tree.
used.
3. Only reflexivity and transitivity of P are


18
by Lemma 2.1. it is unicoherent. Thus aP r\ Pb is also a
continuum. Now, by virtue of the Hausdorff-Maximality-
Principle, aP r\ Pb has a maximal P-chain, C, and
(a) C is closed (Theorem 1.5)> and
(b) C _is connected.
For if C were not connected there would be two non-void
disjoint closed sets A and B such that C = A U B
and b £ B. The set A contains a maximal element m.
Define A' and B; by the equations
A/ = Pm o C and B/ = C Pm.
B C mP, and since A C. A1 then B/ C B. Now
A1 r\ B1 C Pm n (mP r\ B) = (Pm rN mP) r\ B = Q ,
therefore
C = A' U b'
is a separation. If bQ designates the minimal element
in B', then by the maximality of C
mP r\ Pb = { m, b }
o o J
which contradicts the connectedness of mP r\ Pb .
o
Therefore, any maximal P-chain in aP r\ Pb is connected.
(c) aP r\ Pb has a unique maximal P-chain,
which we denote by Cp(a> b).
If C and C' were two distinct maximal P-chains
in aP n Pb, then both C and C/ would contain a and
b, and C U C' would then be connected, and hence C n C'
would be connected. But for x £ C c'


19
C ^ c' = (Px r\ C C) u (xP C r\ C') =
C ^ (Px U XP) r\ C1
Is obviously a separation, a contradiction.
Since (0, b) £ Q(0) C P and since X is a
tree, there is a unique connected Q-chain [22], [23]
Cq(0, b) C Pb which contains both 0 and b. Pb must
also have a connected P-chain containing both 0 and b
and this P-chain must be unique. We denote by C (0, b)
the unique connected P-chain in Pb containing 0 and b.
Since a Q-chain is also a P-chain, then
Cp( 0, b) = CQ(0, b) .
Similarly, there is a unique connected P-chain Cp(0, a)
in Pa containing both 0 and a. It is clear that
Cp(0, a) V Cp(a, b) = Cp(0, b) = CQ(0, b).
As a consequence, a £ Cq(0, b) and hence (a, b) £ Q
which was to be proved.
We next prove the necessity. Let X be a tree
and let P be the outpoint order on X with respect to
a point z in X. We prove that X and P satisfy the
conditions (i), (ii), (iii), (iv) and (v) stated in the
theorem:
Proof of (i). By Ward [24] a tree is a compact
connected commutative idempotent semigroup with zero,
therefore it is acyclic [17]. Hence in particular


20
H^(X) =0. We now show H^A) = 0 for every A = A* C X
and thus X is of one codimension, unless Xis degenerate.
Suppose on the contrary that H (A) ^ 0 for some closed
subset A of X. If h is a non-zero member of H^(A),
then by the Floor Theorem (Theorem l.l6.) there is a floor
F C A for h, which is connected. The set F being a
subcontinuum of a tree is itself a tree and hence is acyclic.
Therefore, H^(F) = 0 which contradicts the fact that F
is a floor, and thus H^(A) = 0. The semilocal connectedness
of X follows from the fact that X is compact and locally
connected.
Proof of (ii). This is proved in Ward [24].
Proof of (iii). The outpoint order P is order
dense [23] and since by (ii) P = P* we have that every
maximal P-chain in Px is connected [22], thus Px is
connected. Indeed, Px itself is a tree and therefore as
has been proved in (i), H^(Px) = 0.
Proof of (iv). This is obvious, since P has
the least element z.
Proof of (v). Replacing Px by xP in the
argument of (iii), we easily obtain the connectivity of xP.
Proof of Lemma 2.3 (l) There is a_ component
Aq of A such that M does not separate Aq and B in
X. For if otherwise, to each component A^ of A there is
a pair of disjoint open sets and T^ such that


X M = U and } A^, T^ ^ B. Since A is
either compact or consisting of finitely many .components,
there is a finite subfamily { G^, G^, .., G \ of G-^s
21
such
that A C
m
v {
Gi 1
m
i =
1, 2, ..
., m } Whence
G =
W G. and
i*l 1
T =
r\
i=l
T.
1
are two
disjoint open sets
such
that X M
= G
KJ t
and
G P A,
T p B so that the
hypothesis (i) is contradicted.
Throughout the rest of the proof, let #[S] be
the number of components of the space S. Let Xf = { | A
be the collection of all open sets containing M but
missing A such that #[X V^] is finite. For each
£ 1/- let R^ be the component of X that contains
the component Aq of A. Designate R ^ u{R^ A e A }
(2) R i_s open. Let y e R, and let £
be such that R^ contains y. By the normality of X
there is a V, e {/ with V C V* C V, Since X V,
[X |X |X A [X
has only a finite number of components and since
(x y0 3 X vj 3 x Vv
we have R, C R C R* Therefore R is open.
A P
(3) For each VQ e ¡j- there is an R^ containing
R such that
a
n (X va)] = #[rx (X vo)]


22
for all containing R^. Let C-^, Cg, .., be the
n components of X V so that
X Va = C-^ o Cg U
where = R^, then for any contained in VQ we have
X containing X Va, so that each Cf must be either
totally contained in R^, the component of X contain
ing Aq, or disjoint from R^. Thus
#[Rp rs (X Va)] < # (X Va) = n.
But,
# [R6 rv (X va) ] < § [RT r\ (x va) ]
if R^ R^ } R Hence there is an R^ } R^ such that
the number #[R^ r\ (X V^)] is the maximal so that
#[RM. for all R^ containing R^.
(4) For each VQ e there is an R^ such that
n ^ (X va> = \ ^ (X va)
for all R^ containing R^. By virtue of (3) above, there
is an R, containing R such'that
A Ct
#t>V ~ (X Va) 3 = #[R* ^ (X Va)]
for all R containing R^. We note, further, that
*V ^ (X Va> \ ^ (X Va)
for all R^ containing R^. For if as in (3) above,
we express
X Va = C1 u C2 u ... u cn
where, without loss of generality, CL is contained in R^


23
for 1 = 1,2, . ., k (k < n) and Ck r\ R^ = O for
i = k + 1, . ., n, then it follows from R .containing
[X
R^ and
#[R,, ^ (X VQ)] = # [R* ^ (X Vj]
cr
that C. r\ R = Q for i = k + 1, ..., n. And therefore,
1 [X
Rp. ^ = C1 u c2 u
= RA ^ u c
k
(3) R = R* M; that is, R _is closed in X M.
If y is a point of R M then there exists a Va in
V- missing y. By the normality of X there is a \k
in If such that C V^* C VQ. By (4) there is an R^
containing R^ such that
A (X Vp) = R* A (X Vp)
for all R^ containing R^. Furthermore, if U is an
open set about y, then U ^ R^ 41 O For, if we designate
the set U r\ (X V^*) by W then y is in both R M
and W so that W ^ R 4= O and there is an R^ such
that W n R =p Without loss of generality we may
assume that R, contains R, Then
o A
D 4= W n R = W ^ Ra C (X Vp)
= W r\ R^ r\ (X Vp)
= W A R .
Thus, U ri R^ } W A R^ 4: so that y is in R^ = R^.
Therefore, R is closed in X M.


It is to be noted that R meets B and hence
R T r\ B ^ Q for some r ; because otherwise.
X M = R U(X-M-R)
would be a separation, of X M between Aq and
that (l) would be contradicted. We now conclude
result by taking N = R^ .
, so
the


CHAPTER III
FIXED POINTS AND MINIMAL ELEMENTS
In the Symposium of General Topology and its
Relations to Modern Analysis and Algebra (Prague 196l),
Professor A. D. Wallace announced [18] among other things
the following fixed point theorem.
Theorem 3-1 [18, Theorem 5]* If X is a
continuum, if P
is a closed
left
monotone partial order
on X such that
pa"* c ( pa)*"
for
each
A C X, and if
z separates Pa
and Pb in
X,
then
Pz = z.
Wallace applied this theorem to prove
Theorem 3 2 [18, Theorem 6]. If X is a con
tinuum and if P is a closed left monotone partial order
on A such that PA*C (PA)* for each A ( X, then the set
K of P-minimal elements is connected.
However, in [18] Wallace gave no proof of the
fixed point theorem. For the sake of completeness, we will
give in this chapter a proof of Theorem 3*1 by first proving
Theorem 3-2 and using this result to establish Theorem 31
We will also show that under certain conditions the state
ments
(i) If z separates Pa and Pb in X then
25


26
Pz = z.
r
(ii) The set of P-minimal elements 'is connected,
are equivalent (Theorem 3**0-
Furthermore, if P is a partial order on the
*??' o
continuum X and K is the P-minimal elements in X,
we will prove in Theorems 3-5^ 3*6 and 3-7 that the
cohomology groups of X and those of K are isomorphic
for all non-negative dimensions for certain classes of X
with suitable choice of P.
The following lemma will be used in the proofs of
foregoing theorems.
Lemma 33 [cf. l6]. If X is a compact Haus-
dorff space and if P is a. lower semi-closed partial order
on X such that PA* < (PA)* for each A ( X, then the set
K of P-minimal elements is closed.'
Proof. Suppose by way of contradiction that there
is an x in K* K. Then since Px is closed and X is
compact, Px has a minimal element which must be a P-minimal
element in X, and thus
Px r\ K \ Q .
Let y be any element in PxaK. Obviously x % y, so
that there is an open set U containing x whose closure
excludes y. If V = U r\ K, then PV = V, and x is in
V*.
Since


27
PV*C (pvf = V*,
then PV* C U* and hence y e Px C PV* C U*. This contra
dicts the fact that y ^ U*. Therefore the set K is
closed.
Proof of Theorem 3*2. Let us assume contrary to
the conclusion of the theorem that K is not connected.
By Lemma 3-3 there are two disjoint non-void closed sets
A and B such that K = A ^ B. Since P is closed, then
AP and BP are closed and X = AP w BP. Furthermore, the
connectedness of X yields AP os BP % Thus AP rs BP
has a minimal element, say t. Since
Pt = (Pt r\ AP) V (Pt BP) ,
then
(Pt AP) r\ (Pt BP) = Pt rs (AP r\ BP) = t.
If we designate
C = (Pt r\ AP) t and D = (Pt/^ BP) t,
then both C and D are open in Pt. Furthermore, C and
D are both non-void. For if C is void, then Pt r\ AP = t,
and so t must be in A. This implies that t is not in
BP which contradicts the fact that t is a minimal element
in AP r\ BP. Similarly, D is not void. Consequently,
Pt t = C U D is a separation. The connectedness of Pt
yields C* = Pt r\ AP and D* = Pt r\ BP. Now, for each x
in Pt t we have Px C Pt t and the connectedness of
Px then implies that either Px C C or Px C D. Thus,


28
for each x in C we see that Px C C, and hence PC C C,
and it follows that
C* 3 (PC)* b PC = P(C ^ t) ) Pt 5 D
which is a contradiction. Therefore K is connected.
Proof of Theorem 3-1- Let X z = A ^ B where
A and B are non-void separated sets and aP ( A and
bP C B. Let K be the set of P-minimal elements. Now,
if Pz ^ z, then K C X z and
K = (K r\ A) V (Kn B)
is a separation for K which contradicts the connectedness
of K.
Theorem 34. If X is a continuum and P is a
closed left monotone partial order on X such that the set
of P-minimal elements K is closed, then the following
statements are equivalent:
(i) If z separates Pa and Pb in X then
Pz = z.
(ii) The set K is connected.
Proof. (i) implies (ii). Suppose the set K is
the union of two disjoint non-void closed sets A and B.
Since K is the set of P-minimal elements then
X = KP = (A ^ B)P = AP U BP.
And the hypotheses that X is connected and P is closed
yield AP r\ BP 4= Let z be a P-minimal element of
AP BP. Then there are elements a e a and b^B such


29
that {a, b}< Pz z. The equalities
Pz = (Pz ^ AP) b (Pz n BP) and
Pz a (AP r\ BP) = z
show that
Pz z = (Pz r\ AP z) ^ (Pz A AP z)
is a separation, that is, z separates a = Pa and b = Pb
in the continuum Pz, and so by (i) z is a minimal element
which is neither in A nor B. This contradiction estab
lishes the connected ness of K.
(ii) implies (i). The proof as given for Theorem
3-1 applies here as well.
We now present an example to motivate Theorem 3*3.
Example. Let X = { (a, b) 1 a2 4- b2 = 1 ^ U
{(0, b) | 1 <_ b <_ 2} for reals a and b. Let X be
endowed with the Euclidean topology. Let
P = { ((0, b1), (0, b2)) [ 1 < bx < b2 < 2j U A
where A = | (x,x) | x £ X } Then P = P* is a both left
and right monotone partial order on X such that PA C (PA)
for each A ( X and
K = { (a, b) l a2 + b2 = l} = S1 (1-sphere)
is the set of P-minimal elements.
It is to be noted in this example that H^Px) = 0
for each x e X while H^(K) ^ H^(S) + 0 for any non
trivial coefficient group. However the following equality


30
Hp(x) = HP(K) holds for all non-negative dimensions. .
It is interesting to seek conditions on X and
a relation P which imply the equality HP(X) HP(K) for
all non-negative dimensions. The purpose of the next
theorem is to take a small step in this direction.
Theorem 3-5- If X is a continuum, and if P
is an upper semiclosed partial order on X such that
(i)PA* C (PA)* for all A C X,
(ii)there exists an element u in X such that
X K C Pu where K = {x| Px = x }, and
(iii)the quotient space X/K x K u A modulo
K X K U A is a topological semilattice under the natural
partial order induced by P, where A { (x, x) I x e X } ,
then, Hp(x) s H^K) for all integers p > 0.
Proof. For simplicity in notation we write
Y = X/KXKUA. Since every compact topological semilattice
has a (unique) zero, we write z for the zero of the semi
lattice Y. By the hypothesis (ii), Y has a (unique) unit
which will be denoted by u also since no confusion is
likely to occur.
We will accomplish the proof in three steps.
(a) H^(X, K) = HP(Y, z) for all integers p >_ 0.
Let f : (X, K) -> (Y, z) be the natural map. Since (X, K)
is a compact pair, f is a closed map, and moreover, f


31
takes X K topologically onto Y z. Thus-by the Map
r *
Excision Theorem (Theorem 1.13) f*- : HP(Y, z) -> HP(X, K)
is an isomorphism.
(b) HP(X, K) = 0 for all integers p > 0.
We
establish this by showing HP(Y, z) = 0 for all integers
p >_ 0. Define ffc : (Y,z) -> (Y,z) by f (y) = t a y for
all t e Y, where a is the semilattice operation on Y.
If for each t £ Y define : (Y, z) -> (YX Y, z xY) by
At(y) = (y, t) then f^_ = A o Since by the Homotopy
Lemma (Theorem 1.14) 7\* = Au*, thus
f *= A*. A*= A*. A*= f *.
z z u u
Since f^ is the identity map, so is fz. Let
i : (z,z) C (Y, z) be an inclusion map and let h : (Y, z)
(z, z) be defined by h(y) = z for all y e Y. Since
f = i h, the following diagram
Cj
is analytic, for all non-negative integers p, that is
fj = f*= h** i*. Since Hp(z, z) = 0 for all integers
p > 0 and HP(Y, z) = fJ(Hp(Y, z)) C h*(Hp(z, z)) for all


32
integers p >_ 0, then HP(Y, z) = 0 for all integers
p >_ 0 as desired.
(c) HP(X) = H^K) for all integers p >_ 0. By
combining the above results with the exact sequence for
the triple (X, K, ) (Theorem 1.7)* we obtain the exact
sequence
0 = HP(X, K) -> HP(x) > HP(K) -> HP+ 1(X, K) = 0
for p = 0, 1, 2, ... Consequently, Hp(x) and HP(K)
are isomorphic for all non-negative dimensions.
We now center our attention on the case in which
X is of codimension one.
Theorem 36. If P is a closed partial order on
a continuum X and if ir : X^X/KXKUA designates the
natural map, such that
(i)PA*C (PA)* for all A C X,
(ii)P is left monotone and H^(7r(Px)) = 0 for
all x e x,
(iii)X is of codimension one,
then Hp(x) HP(K) for all p > 0.
Proof. Denote X = X/KX A and
P = { (tt(x), ir(y)) J (x, y) p J
then since X is compact and ir continuous, P is closed
and left monotone; indeed we have P7r(x) = tt(Px) By
virtue of Theorem 1.17* we have HP(X, K) = HP(X)
for all


33
p >_ l. Since X is of codimension one we hav. HP(X, K) =
0 for p >_ 2 [21] We show H^X, K) = 0 by proving
H^(X) = 0 Using the same argument employed in the proof
of Lemma 2.2 one sees that H^(PS) = 0 for all closed sets
"I ^
S in X; in particular we have H (X) = 0 We have noted
in part (c) of the proof for Theorem 3*5 that if HP(X, K)
= 0 for all p > 1 then HP(X) s HP(K) for all p > 1.
The equality H^(x) = H(K) follows from the fact that
both X and K are connected [Theorem 3*2].
We remark that the hypothesis (iii) in Theorem
3-6 may be weakened to :
(iii ) cd (X K)* = 1,
and this may even be dropped completely if each Px is a
chain, as will be seen in the following
Theorem 3 7 If P is a closed partial order on
a continuum X such that
(i) PA* C (PA)*, and
(ii) each Px is a connected chain,
then Hp(x) £ HP(K) for all p > 0.
Proof. Let X, 7r and P be defined as in
Theorem 3-6. Since P is closed and 7rPx is a connected
chain P is closed and Ptt(x) is a connected chain. It
follows then that each P7r(x) is a generalized arc and
hence Hp(P7t(x)) = 0 for all p >_ 1. As it has been noted


3^
in the proof of Theorem 3*6, H1(PS) = 0 for 11 closed
sets S in X. We now show H^(X) = 0 for all p >_ 1
by proving HP(PS) = 0 for all p >_ 1 and for each closed
set S in X. If there were a least integer n such that
Hn(PA) i= 0 for some closed set A in X, then n would
have to be greater than 1, and A =f= Let h be a non
zero member of Hn(PA), then h | Pa = 0 for each a e A
and hence by the Reduction Theorem there is an open set V
containing Pa such that h I V =0; then by Theorem 1.1
there is an open set U containing a such that PU C V.
Thus the collection a of all open subsets U of A such
that h J PU = 0 forms an open cover of A. Also Q. is
closed under finite union, for if and Ug are in a
denote hQ = h | PU* U PU*. Since P^ U U2)* = P(U* ^ U*)
= PU-^ V PU2 to show GL is closed under finite union it
suffices to show hQ = 0. In the following part of Mayer-
Vietoris exact sequence :
Hn-1(PU*^ PU*) * Hn(PU*^PU*) Hn (PU *) x Hn(PU*)
since J*(hQ) = (h I PU*, h I PU*) = (0, 0) hQ is in the
image of A. But Hn_1(PU*^ PU*) = Hn_1(PS) where S =
PU*r\?uJf is a closed subset of X, and hence Hn-'L(PS) =
0 by the minimality of n. This proves hQ = 0 so that
a is closed under finite union. Since A is compact,
A is a union of some finitely many elements of (X Thus
A must be in (X and so h = 0, a contradiction. This


35
together with Theorem 1.17 implies HP(X, K) = Hp(x) = 0. for
all p >_ 1. It then follows from the exact sequence for the
triple (X, K, O) [Theorem 1.7] and the connectedness of X
and K [Theorem 3-2] that Hp(x) = HP(K) for all p > 0.
Remark. Theorem 3-7 may be stated more generally
by repacing the hypothesis (ii) by
(ii') each 7r(Px) is acyclic.
We conclude this chapter by exhibiting an example
which answers a question in Topological Semigroups. A clan
is a compact connected topological semigroup with unit [20].
Let S be a clan and let
R = { (x, y) | (x, y) e S X S, xS uSx c yS u Sy }
then R is a closed quasi-order on S and the set of
R-minimal elements is also the minimal ideal of S. The
question to be answered is: if. a. clan S is. a. tree, is
its minimal ideal an arc or a. point 9
The answer is affirmative if S is abelian (or
normal: xS = Sx for all x e S), but it is negative in
general.
Example. Let S = X ^ I be the subset of
Euclidean 3-space such that
X = { (x,0,0) | -1 < x < 1 } U {(0,y,0) | -1 < y < 1}
I = { (0,0,t) f 0 < t < 1 } ;


36
let S be endowed with the
Euclidean topology and let
the multiplication o on
S be defined below (the
usual multiplication of
reals is denoted by
juxtaposition) :
i. Aop =
ii. (I, o)
unit interval; i.e.,
A for all A £ X and for all p e S
is the usual semigroup of the real
(0,0,tx) o(0;0,t2) = ( 0, 0, t-^g) ,
iii. For each (0,0,t) e I and for any (0,y,0)ex,
(0,0,t)o(x,0,0) = (tx,0,0) and (0,0,t)o(0,y,0) = (0,ty,0).
Then, (l) (S, o) is associative.
(2) o : S S -> S is continuous.
(3) (S, o) is a clan with (0,0,l) as unit.
(4) The minimal ideal of (S, o) is X which
is neither an arc nor a point.
It is interesting to observe that the semigroup
S given in the above example may be realized as a semi
group of matrices by the following one-to-one correspond
ence :

t
x + yf-1
(x,y,t) +
0
1


37
where -l<_x, y <_ 1, 0 < t < 1 and (tx)2 + (:xy)2 + (yt)2
= 0. The correspondence is indeed an isomorphism. Prom
this the assertions (l), (2) and (3) in the example
are self-evident.
/


CHAPTER IV
FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES
A space X is said to have th _f. jd. jd. (fixed
point property) if, for every continuous function f: X -> X
there exists some x in X such that x = f(x). Hamilton
[8] has proved that the chainable metric continua have the
f. p p .
A space X is said to have the F. jd. jd. (fixed
point property for multifunctions) if every continuous
multifunction'*' F : X -> X has a fixed point, i.e., there
exists a point x in S such that x e F(x). Obviously
if X has the F. p. p. then it has the f. p. p., but
the converse need not be true. Strother [13] has exhibited
two continua X and Y both have the F. p. p. but their
Cartesian product X xY fails to have the F. p. p. Borsuk
[3] has constructed a decreasing sequence of three-cells
whose intersection does not have the f. p. p. The inter
section is the inverse limit, the bonding maps being in
jections. As a counter theorem to the results of Strother
and of Borsuk, we prove that if (X-^, ir^ /\ ) is an
1. Following Strother [12], a multifunction
F : X -* Y is continuous if, and only if, F(x) is closed for
each x, and F ^(A) is open (closed) if A is open (closed).
38


39
inverse system of compact spaces such that each X^ has
the F. p. p. then the inverse limit space has the F. p. p.
As a corollary to this, we obtain Ward's generalization [25]
of the Hamilton theorem [8] that every chainable metric con
tinuum has the F. p. p. Our result is, indeed, stronger
than the Ward's, since it includes some of the non-metric
chainable continua as well.
Definition 4.1. The collection (X-^, tt^ -A )
is as inverse system of spaces if:
(i)A is a directed set,
(ii)A in A implies that X^ is a Hausdorff
space,
(iii)whenever A > p. there is a continuous
function ir-. : X, -> X, ,
Ap, A p.
(iv)if A > p. and p, > v, then tt^ =
The function v^ is called a bonding map. If A
is in A let be the subset of the Cartesian product
P{ X^ | A e A } defined by
| x | if A > p. then tt-^x(A) = x(p.)} ,
where x(A) denotes the A-th coordinate of x.
Definition 4.2. The inverse' limit space Xq, of
the inverse system of spaces (X-^, tt^ A ) is defined to
be
= r\ {
A
A e A }


40
endowed with the relative topology inherited fpom the
product topology for P { | A e A } ; in notation X
- J (XA- A )
We write p^ : P { X^ | A e A } * X^ for the A-th
projection of P [ X^ | A e A } i.e., p^(x) = x(A) for all
xin P|X^| A^A}; the restriction p^ j X^, will be de
noted by 7which will be called a projection map. It is
readily seen from the definition that an element x of
P { XA | A e A] is in X^ if and only if tt^^tt^x) = tt (x)
whenever A > p. A more detailed account of inverse limit
space may be found in Lefschetz [10], Eilenberg and Steenrod
[7], Capel [4] and Mardesic [11].
The following known results (see, e.g., [4], [10])
will be used.
Lemma 4.3 (i) The collection ¡ A £ A
and is an open subset of X^ j forms a basis for the
topology of X.
(ii)The inverse limit space X^, is Hausdorff;
if A e A is a closed subset of P { X^ [ A e A} so
that X^ is closed in P { X^ | A e A }
(iii)If X^ is compact for each A in A then
X^ is compact; if, in addition, each X^ is non-void
then X is non-void.
(iv)If X^ is a continuum for each A ^A then
the inverse limit space is a continuum.


Lemma 4.4. If A is a compact subset of X^
and if 7= ir^ | then (tt^(A), ir^ A) is an
inverse system of spaces such that A = 11m (tta(a), tt^, A)
and each bonding map tt^ is onto.
In the sequel, since we are only interested in
compact spaces, each projection map twill be assumed
to_ be_ onto; for if otherwise, by virtue of Lemma 4.4,
each may be replaced by tt^X*.) without disturbing
the resulting inverse limit space. We are now ready to
state our main result.
Theorem 4.5. Let (X^, tta A) be an inverse
system of compact spaces such that each X^ has the P. p.
p., then the inverse limit space X*, also has the F. p. p.
We divide the proof of this theorem into the
following steps. In Lemmas 4.6, 4.7 and 4.8 X,*, will be
assumed to be the inverse limit space of the inverse system
(X^, tt^, A ) of compact spaces.
Lemma 4.6. If F : X# -> X^ is a continuous
multifunction, define F^ : X^ -> X^ by F^ = m^F tt^ for
each A. Then F, is a continuous multifunction.
A
Proof. (i) If t is in X^, then since F : X^ -* X
is a continuous multifunction, by Theorem 1.1 and Theorem
1.4 of Chapter I, Fm^^t) is a closed subset of X^. Thus,


F^it)
is closed for every
t in
X, .
(ii) If CA is a
closed
subset of XA, then
is closed. It is
readily
seen that F ^tta^(Ca)
Is closed in Xm and hence compact; therefore
,ir^F~'Sr^ (C^) = F^(C^) is compact and hence closed.
(iii) If UA is open in XA, then F^(U^) is open.
7rA and F being continuous, F 'Tr-^(U^) is an open subset
of Xoo It follows then, by virtue of Lemma 4.3 (i), that
7r^F_17T^1(U^) = F^(U^) is open.
Thus, by (i), (ii) and (iii) above, F^: X-^X-^
is continuous.
Lemma 4.7 Let F : Xffl -> X, be a continuous
multifunction, let F^ : X^ -> X^ be defined as in Lemma 4.6
Then, for each x in Xa, ,
(i) (F^ir^(x) ttA[X, A)2 and (ttaF(x) ttA[x, A )
are inverse systems of compact spaces,
(ii) lim (FA7rA(x), 7tA[x, A) = lim (ttaF(x) ^,A)
(iii) F(x) = lim (Fa7Ta(x), 7Ta^, A).
Proof. Since each FA is continuous (Lemma 4.6)
and each XA is compact, so is for all A e A .
2. For simplicity in symbolism, henceforth if
A C lim (XA, 7r A^, A) then (u^A, tta A) will mean
KA a )


43
To show that (F^tt-^(x) tt^ A ) forms ah inverse system,
it suffices to show 7T^^F^tt^(x) £ F^Tr^(x) whenever A > \i.
To this end we first observe
Vx> < h^irAlx)^(x) = ir^fx),
sinc 7T-, TT-> = 7T From this we have
Ap A [X
^AixVa^) C rAM.VV(x)
= VV r1)rV(x)
= V v\(x)
= F 7T (x) ,
(1 [l
by the definition of F^, F^ and the equality 7r^7r^ =
The fact that (7r^F(x) tt-^, A) forms an inverse
system follows from Lemma 4.4.
(ii) For each A e A and any x e X^, we have
x e Tr^7r^(x) and hence,
n\F(x) C tt-vF tt'VAx) = (tt.F v~)rli(x) = F,7iu(x)
-1>
Thus,
lim (7TaF(x), tt^, A)c^ (F^U), tt^, A).
To prove the other inclusion, we show
Xoo lim (ttaF(x), ttA[i, A) C Xm- lim (F^tt^x) ir^, A).
Let y be in Xa lim (ttaF(x), /\ ) then, by Lemma 4.4


44
there exists a p e A such that ir^(y) tt^P(x) Let
and V be two disjoint open sets in X such that
P M-
Vy) £ UH and V(x) C VH
so that
F c
It follows then from Lemma 4.3 (i) and the continuity of
F that there exists a 6 £ A and an open set in X^
such that x a ir^U^ and
(*> fKX) c y\
Since A is directed, there is a Aq e
A > ll and A > 6, we shall use this A
o o
proof of this lemma. If we write U, =
Ao
the equality Tr^1^ then ( )
A such that
throughout the
n\"\Uc and use
A qO u
may be rewritten
as
and hence
C T-\ ,
F, U, = 7T, F 7T,1(U, ) / 7T, 7T 1V = 7T, (iT, TT, ) 1V
Aq Aq Ao A0 Aq V Aq P P A0 AqP Aq p
In particular,
F, TT, (x) / IT, 1 V .
Aq Aq V A0p, |1
Similarly, one obtains
(y) 6
since tt (y) £ U and v = 7r, 77\ The fact that U
P P P Aq p
are disjoint implies '^AqI.^ij,rv'n'AoP^p = ^ and
and


45
consequently 7r, (y) 4 F, tj\ (x) From this we conclude
Ao Ao Ao
y ^ lim (F^tt^(x), -it^ A ) as desired.
(iii) This follows immediately from (ii) and
Lemma 4.4.
Lemma 4.8. Let F : X^, be a continuous multi
function, let F^ : X, -> X^ be defined as in Lemma 4.6 Let
EA = {eA I eA 6 XA ancl eA e VeA>} fchen
forms an inverse system.
Proof. It suffices to prove 7r^(_L41^ £ E^ whenever
A > p. Let e^ e F-^(e^), then
£ = Va^A^A* = VV
c V(V1rH)r1(eA) : FnV1(eA)
= WaK^a* = ¡'u(1rA|X(eA))-
-1,
Thus, ir^E^ C E^ as as to be Proven.
Proof of Theorem 4.5 Since each X^ has the
F. p. p. and by Lemma 4.6 each F^ : X^ - X-^ is continu
ous, each E^ is closed and non-void. By Lemma 4.8,
(E^, tt^ A ) is an inverse system of compact spaces, so
it has a non-void inverse limit space lim (E-^, tt^, A ) .
We now conclude the proof by showing that each x in
(Ea, ttA/U, A ) is a fixed point under F; i.e., x£F(x).


46
If x is in lim (Ev 7rA^, A) then tta(x) for-all
A G A ; i.e., tta(x) e for all A G A Conse
quently, by Lemmas 4.4 and 4.7, we have
x = lim (irA(x), tt^, A) g lim (P^(x) vA)
= F(x).
In fact, with the assumption of Theorem 4*5 and
the notation of Lemma 4.8 together with the notation
E = | x | x G F(x) J. we can make the following sharper
assertion.
Theorem 4.5* E = lim (E^, tt^, A ) .
Proof. Prom the proof of Theorem 4.5, we have
E ) lim (Ea, tt^, A) .
It remains to prove that
E C lim (Ea, tt^, A )
Let x be in E, then x G F(x) and therefore, for all
A e A ,
tt^(x) g tt^F(x) C tTaFCit-VaJCx) = F-Jtt^x)).
That is, tta(x) e for all A; consequently, by Lemma 4.4
E < lim (Ea, ir^ A )
A chain (U^, U^, , U ) is a finite sequence
of sets U. such that U. r\ U. if and only if
J
| i j i < 1. A Hausdorff space X is said to be chainable


47
.if to each open cover of X there is a finite open
cover /[X = (U^, Ug, .., Un) such that (i) U refines
V"> (ii) u= ( Uf, Ug, .., Un) forms a chain. It
follows that a chainable space is a continuum. It is
implicit in a paper by Isbell [9] that each metrizable
chainable continuum is the inverse limit space of a sequence
of (real) arcs. This together with a theorem of Strother
[13] that a bounded closed interval of the real numbers has
the F. p. p. imply the following result of Ward [23] as a
consequence of Theorem 4.5-
Corollary 4.9. Each chainable metric continuum
has the-F. p. p.
Examples of inverse limit spaces of inverse
systems of real arcs exist which are not metrizable; for
instance, the "long line" is one such. Thus, Theorem 4.5
is a proper generalization of that of Ward's [25]*


BIBLIOGRAPHY
1. Anderson, L. W. and Ward, L. E., Jr. One-dimensional
topological semilattices. Ill. J. Math., 5
(1961), 182-186. ,
2.
Borsuk, K. lber die Abbildungen
Raume auf die Kreislinie.
der metrischen kompakten
Fund. Math., 20 (1933),
224-231.
3. Sur un continu acyclique qui se laisse trans
former topologiquement en lui meme sans points
invariants. Fund. Math., 24 (1935), 51-58.
4. Capel, C. E. Inverse limit spaces. Duke Math. J., 21
(1954), 233-245.
5. Cech, E. Sur les continus Peaniens unicoherents. Fund.
Math., 20 (1933), 232-243-
6. Cohen, H. A cohomological definition of dimension for
locally compact Hausdorff. spaces Duke Math. J. ,
21 (1954), 209-224.
7* Eilenberg, S. and Steenrod, N. Foundations of algebraic
topology. Princeton University Press, Princeton
1952.
8. Hamilton, 0. H. A fixed point theorem for pseudo-arcs
and certain other metric continua. Proc. Amer.
Math. Soc., 2 (l95l), 173-174.


9.
10.
11.
/
12.
14.
Isbell, J. R. Embeddings of Inverse limits 4 Ann. of
Math., 70 (1959), 73-84.
Lefschetz, S, Algebraic topology. Amer. Math. Soc.
Colloq. Publ., no. 27, New York 1942.
Mardesic, S. On Inverse limits of compact spaces.
Glasnik Mat. Flz. Astr., 13 (1958), 249-255-
Strother, W. L. Continuity for multi-valued functions
and some applications to topology. Doctoral
Dissertation, Tulane University 1952.
On ari open question concerning fixed points.
Proc. Amer. Math. Soc., 4 (1953), 988-993-
Wallace, A. D. A fixed point theorem. Bull. Amer. Math
Soc., 51 (1945), 613-616.
15- Cohomology, dimension and mobs. Summa
Brasil. Math., 3 (1953), 43-55-
l6. Struct ideals. Proc. Amer. Math. Soc., 6
(1955), 634-638.
17 Acyclicity of compact connected semigroups.
Fund. Math., 50 (l96l), 99-105-
18. Relations on topological spaces. Proc. Symp
on General Topology and its Relations to Modern
Analysis and Algebra. Prague 1961, 356-360.
19. Relation-theory, Lecture Notes. University
of Florida, 1963-1964.
20. Topological semigroups, Lecture Notes.
University of Florida, 1964-1965-


50
21. Wallace, A. D. Algebraic topology, Lecture Notes.
University of Florida, 1964-65
22. Ward, L. E. Jr. Partially ordered topological spaces.
Proc. Amer. Math. Soc., 5 (1954), l44-l6l.
23- _A note on dendrites and trees. Proc. Amer.
Math. Soc., 5 (1954), 992-994.
24. Mobs, trees, fixed points. Proc. Amer.
Math. Soc., 8 (1957), 798-804.
25- A_ fixed point theorem. Amer. Math. Monthly,
65 (1958), 271-272.
26. Whyburn, G. T. Analytic topology. Amer. Math. Soc.,
1942.


BIOGRAPHICAL SKETCH
Shwu-yeng Tzeng Lin was born May 11, 193^ at
Tainan, Formosa. In June, 1953, she was graduated from
Tainan Girl's High School. In June, 1958> she received
the degree of Bachelor of Sciences from Taiwan Normal
University. She worked as a Research Assistant at the
Mathematics Institute of Academia Sinica from 1958 to
I960. In the fall of I960 she enrolled, and worked as a
teaching assistant, in the Department of Mathematics,
Graduate School of the Tulane University, New Orleans,
Louisiana, and received the degree of Master of Sciences
in May, 1963- From September, 1963, until the present
time she has pursued her work toward the degree of Doctor
of Philosophy at the University of Florida.
Shwu-yeng Tzeng Lin is married to You-Feng Lin
and is the mother of one child.
51


This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Arts and
Sciences and to the Graduate Council, and was approved as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
August, 1965
Dean, College of Arts
and Sciences
£7, &


UNIVERSITY OF FLORIDA
V ^ 'i'1""",,,, I'll III'
3 1262 08556 7369






Full Text
£P£ S6 (9. t£ 3 3-P3~\ i
36 relationsonspaceOOlins
i



RELATIONS ON SPACES
By
SHWU-YENG TZENG LIN
A 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
August, 1965

ACKNOWLEDGMENTS
The author wishes to acknowledge her indebtedness
and express her sincere gratitude to Dr. A. D. Wallace,
Professor of the Department of Mathematics and Chairman
of her Supervisory Committee, for his guidance and sugges¬
tions throughout the preparation of this dissertation.
She wishes to express her deep appreciation to Professor
A. R. Bednarek, who read the entire manuscript in detail
and made numerous corrections and improvements. Thanks
are due also to all members of the Supervisory Committee
for their encouragements.
The author is grateful to Dr. D. C. Rose for
correcting the language. She is also grateful to her
husband Dr. Y.-F. Lin, who was always patient in discus¬
sing mathematics with her and who rendered many valuable
suggestions. And she extends her thanks to Mrs. K. P.
Grady for her excellent work in typing the manuscript.
ii

TABLE OP CONTENTS
Page
ACKNOWLEDGMENTS ii
Chapter
I.PRELIMINARIES 1
II.A CHARACTERIZATION OF THE OUTPOINT
ORDER ON A TREE 11
III.FIXED POINTS AND MINIMAL ELEMENTS 25
IV.FIXED POINT PROPERTIES AND INVERSE
LIMIT SPACES 38
BIBLIOGRAPHY 48
BIOGRAPHICAL SKETCH 51
iii

CHAPTER I
PRELIMINARIES
A relation R on a topological space X is_ a_
subset of X x X. If A is a subset of X and if R is
a relation on X, let
AR = p2 ((AXX)^R)
and
RA = P]_ ((X x a) r\ R)
where p^, p2 are respectively the first and the second
projections. For x in X we shall write simply xR for
{ x } R.
Let
R
(-D _
= { (y, x) | (x, y) e R } ,
and
AR^ { x | xR C A } ,
R['1]A= {y | Ry C A } .
In all that follows, X will always denote a Hausdorff
space, and R a relation on X. The closure of a set A
will be denoted by A* and the interior of A by A° .
Definition 1.1. A relation R on X is
said to be
lower semi-continuous (abbreviated l.s.c.)
upper semi-continuous (abbreviated u.s.c.)
1

2
at x in X if and only if x in
(vr(-1))° >
l for all V = Vo C X. R is said to be
(W 0 J
.l.s.c. ,
on X if and only if R is
,l.S.C* \
at
u.s.c. /
* U.S.C. i
x for all x in X. R is said to be continuous if and only
if R is both l.s.c. and u.s.c.
Definition 1.2. A relation R on X is
, upper
semiclosed
(u. s .cl.) -j
V if and only if
n
\ lower
semiclosed
(l.s.cl.)J
^ Rx J
is closed
for any x
in X.
Definition
1.3* A relation R on X
is said to
be a partial order1 if and only if the following conditions
are satisfied:
(a) (reflexivity): (x, x) £ R for all x in X,
(b) (antisymmetry): (x, y) 6 R and (y, x) £ R
imply x = y,
(c) (transitivity): (x, y) 6 R and (y, z) £ R
imply (x, z) £ R.
VR
â–  VR
(-D
[-1]
â–º implies x in
1. A relation R is said to be a quasi-order if
(a) and (c) are satisfied.

3
We here state some theorems to which we will refer
throughout this work. The proofs of these theorems may be
found in [19].
Theorem 1.1. If R is closed in X x x and if
A is compact, then both AR and RA are closed; moreover,
if V is an open set then R^-^ V and VR^-1^ are open.
Theorem 1.2. The following statements are equi¬
valent :
(a) R is l.s.c. on X.
(b) RA° £ (RA)° for all A C X.
(c) A* R C (AR) * for all AC X.
(d) RV is open for all open VC X.
(e) p^ | R is open, where p^ | R is the
restriction of p^ to R.
Theorem 1-3- The following statements are equi¬
valent :
(a) R is u.s.c.
(b) VR^"1^ is open for all V C X.
(c) AR^-1) is closed for all closed A C X.
(d) A° R^"1^ C (AR[_1])° for any A C X.
(e) (AR^_1^)^C A* r(-1) for any A C X.
Theorem 1.4. If X is a compact Hausdorff space,

4
then R = R* if and only if
(a) R is u.s.c. on X, and
(b) R is u.s.cl. on X.
Definition 1.4. A subset C of X is an R-chain
if and only if C x c C R U R^-1^.
As a consequence of the well-known Hausdorff
maximality principle, there is a maximal R-chain for any
relation R on any space X. A useful result of
Wallace [14] is the following.
Theorem 1.5- If R is both u.s.cl. and l.s.cl.
quasi-order, then every maximal R-chain is closed.
Definition 1.5- An element a in X is R-minimal
(R-maximal), whenever (x, a) e R ((a, x) e R) implies
(a, x) e R ((x, a) 6 R).
The following fundamental theorem first proved in
[14] will be used repeatedly.
Theorem 1.6. If R is a u.s.cl. (l.s.cl.)
relation on a compact space X, and if A is a non-void
closed subset of X, then A contains an R-maximal
(R-minimal) element a of A.

The Alexander-Kolmogoroff cohomology groups will
he used as developed in [21]. In what follows the coef¬
ficient group is fixed and therefore will not be mentioned.
We record here some useful theorems from [21].
Theorem 1.7. If B C A C X, then
H°(X, A) -X H°(X,B) H°(A,B) — >
H1(X, A) —¿->H1(X,B) —^>H1(A,B) ——*
6
^
Hn(X, A) —J >
is an exact sequence, where i* and j* are the induced
homomorphisms of the inclusion maps i and j from A to
X and X to X, respectively, and 6 is the coboundary
operator for the triple (X, A, B).
Theorem 1.8. For the space X and any connected
set A of X the homomorphism i* of
H°(X) -X H°( A) X H1(X, A)
is an epimorphism and hence 6 is 0.
For simplicity we denote by f:(X, A) -» (Y, B)
that f is_ a_ function from X to_ Y and A is. Ü subset
of X such that f(A) C B C Y. Furthermore, if
i:(X, A) -> (X, B) is_ an inclusion map, we will write

6
i:(X, A) C (X, B).
Theorem 1.9.
If
f
:(X, A,
B)
continuous, and if u:
(x,
A)
(x',
A')
v:
(x,
B)
-> (X',
B')
w:
(A,
B)
â– > (A',
B')
are defined by
u(x) = v(x) = w(x) = f(x),
then the ladder
———»HP (X, A) —¿->HP(X, B) -^%HP(A, B) - 5 y
—5~>HP(X, A)—^HP(X, B) tt^HP( A, B)—g->-
is analytic, that is, each rectangle of the ladder is
analytic.
We state here a special case of The Mayer-Vietoris
Sequence which will be sufficient in what follows for our
purpose.
Theorem 1.10. If X is a compact Hausdorff space,
and if X = X-^ Xg where X^ and Xg are closed subsets
of X, then there exists an exact sequence
—A Hq_1(X1 r\ X2) -A> Hq(X) -iX Hq(X1) X Hq(Xg) —
Hq(xx r\ x2) -A* ...

7
where (i* , i* , ó ) and (j * , i* , ó ) are the homo-
v °a ’ a ’ or ' a ’ or
morphisms in the exact sequences for the triples (X, X , â–¡ )
and
(Xa, Xx X2
, a
)
respectively for
k:
(X2, X1 kh X2)
c
(x,
x1) ; J* = x
I*
- 1 1 1 2 ’
and
A
k ó2 ‘
As a consequence of the foregoing theorem we have
the following
Theorem 1.11. With the hypotheses of Theorem
1.10 and if X-^ r\ Xg is connected, the homomorphism A in
h°(x1 r\ x2) -A* h1(x) -A_> h1(x1) X h1(x2)
is 0.
Proof. Observe the following ladder
H°(x) 4 H1(X, Xx) —^ H1(X) — >
u* k*
'r -g- -r* -t-'H
H°(X1 ^ X2) —%■ H1(X2, X1 ^ X2) % HX(X2) —^
where u: Xx a X? C X and where (j*, i*, ¿1) / (J*, Tg, ó^)
and k are as in Theorem 1.10. It follows from Theorem 1.8
that if h is in H°(X^ f\ Xg) then there is an h' in
H°(X) such that u*(h') = h. Thus by Theorem 1.9., we have

8
k* _1 óg(h) = k* k* 61(h/ )
» • •
which contains ¿1(h/). And A = j k* 6^ being
a well-defined homomorphism, we see that for any a and 8
in k * -1 6*2(h) , j*(a) = j*(8). Therefore,
A(h) = j* k* _1 02(h) = j*<51(h/ ) = 0.
The following notation is convenient. _If P
is a_ subset of Q and if h e Hp(Q) then h | P denotes
the image of h under the natural homomorphism induced by
the inclusion map of P into Q.
Theorem 1.12. (Reduction Theorem). If
compact Hausdorff space, if A is closed, and if
such that h | A = 0, then there is an open set U
A such that h | U =0.
Definition 1.6. (X, A) is a compact pair if and
only if X is a compact Hausdorff space and A is a closed
subset of X.
X is a
h e HP(X)
containing
Theorem 1.13» (Map Excision Theorem). If (X, A)
and (Y, B) are compact pairs, and if
f: (X, A) + (Y, B)
is a closed map such that f takes X - A topologically
onto Y - B, then

9
f* : HP(Y, B) = HP(X, A).
x •
t •
Theorem 1.14. (Homotopy Lemma). If (X, A) is
a compact pair and if T is a connected space, and if for
each t in T
At : (X, A) -> (X x t, A x t)
is defined by A^(x, t) = (x, t),
then A * = A"'*' for r and s any elements of T.
I? s
Definition 1.7» A space X is unicoherent if
and only if X is connected and X = A B with A and
B closed and connected implies A rs B is connected. X
2
is hereditarily unicoherent if every subcontinuum of X
is unicoherent.
Theorem 1.15- If X is a continuum and if
H1(X) = 0 for G £ 0, then X is unicoherent.
Definition 1.8. If X is a space, if A C X
and if h is a non-zero member of HP(A), then a closed
set P C A is a floor for h if and only if h | F M
while h | F' = 0 for any closed proper subset F' of F.
space.
2. A continuum is a compact connected Hausdorff

Theorem 1.16. (Floor Theorem). If (X, A)
is a compact pair and If h is a non-zero member of
HP(A), then h has a floor. Moreover, every floor is
connected.
Theorem 1.17• If (X, A) is a compact pair, if
R = A x A U A where A= { (x, x) [ x e X } then
HP(X, A) es HP(X/R) for all p> 1.

CHAPTER II
A CHARACTERIZATION OF THE OUTPOINT ORDER
ON A TREE
A tree is a continuum such that every two
distinct points are separated by the omission of a third
point. Let X be a tree and let z be an arbitrary,
but fixed element of X. Let Q(z) be the set of all
such pairs (a, b) in X * X such that at least one of
the following three conditions is satisfied :
(i) a » z,
(ii) a = b, or
(iii) a separates z and b in X.
It turns out that Q(z) is a continuous partial order on
X, and with respect to this partial order z is the unique
minimal element. We shall refer to Q(z), for any z in
X, as a cutpoint-order [l] on the tree X. The purpose
of this chapter is to give a characterization of the cut-
point order on a tree (Theorem 2.4). We also obtain a
new characterization of a tree from Relation-theoretic and
Cohomological view-points.
11

12
Several characterizations of a tree have been
given [2], [5], [23], and [24]. Perhaps the most useful
of these characterizations is
Lemma 2.1. [2], [5]- A continuum X is a tree
if and only if it is locally connected and hereditarily
unicoherent.
An excellent proof of this lemma may be found
in Ward [24].
Definition 2.1. A space X is said to be
semi-locally-connected (abbreviated s.l.c.) at a point x
of X provided for any open set U in X containing x
there exists an open set V containing x such that
V C U and that X - V has only a finite number of
components. If X is s.l.c. at each of its points, it is
said to be s.l.c.
In 1933, A. D. Wallace [15] proved that one-
cod imensional"*" compact connected and locally connected
topological semigroup with unit and zero is a tree.
L. W. Anderson and L. E. Ward, Jr. in 1961 [l] modified
Wallace's result by eliminating the necessity of hypothe¬
sizing a unit. More precisely, they proved that if
1. For the definition and properties of Codimension,
see Wallace [21], or Cohen [6].

13
X is a compact connected, locally connected, one-
codimensional topological semilattice, then X. is a tree.
Wallace [17] improved this result by weakening the local
connectedness of X to semilocal connectedness of X.
These elegant results on Topological Algebra, motivated
the following Lemma which bears a Relation-theoretic
analogy.
Lemma 2.2. If R is a relation on a compact
Hausdorff space X with RX a one-codimensional semi-
locally connected subspace, such that
/ \ *
(i)the relation R is closed, i.e., R = R ,
(ii)H^Rx) = 0 for every x in X,
(iii)the collection j Rx | x e X^ has the finite
intersection property (abbreviated f.i.p.), and
(iv)Ra r\ Rb is connected for each pair a, b
in X,
then RS is a tree for every closed subset S
of X.
The proof of Lemma 2.2. depends on the following:
Lemma 2.3- If A, M and B are disjoint non-void
closed subsets of a normal space X, and if A is either
compact or consisting of finitely many components such that
(i) M does not separate A and B in X, and (ii) for

14
any open set U containing M there is an open subset V
of U containing M such that X - V has only a finite
number of components, then there exists a closed and
connected subset N of X such that N <, X - M and N
meets both A and B.
This lemma was first proved by G. T. Whyburn [26]
for the particular case in which X was assumed to be a
metric continuum and card A = card M = card B = 1. The
non-metric case was implicit in a paper by Wallace [17]
but without proof. We postpone the proof of Lemma 2.3* to
the end of this chapter.
Proof of Lemma 2.2. It follows from (i) and
Theorem 1.1. that RX is closed and from (iii) and (iv)
that
RX = J (Rx |x e X)
is connected and thus RX is a continuum. Similarly RS
is a continuum.
Since every subcontinuum of a tree is itself a
tree and since RS is a subcontinuum of RX, it is
sufficient to show that RX is a tree.
We first show H^(RX) =0. If there were a
non-zero h e H1(RX), then there would be a maximal (non¬
void) tower J of closed subsets A of X such that
h i RA f 0. Let Aq= n { A | A £ [[ )
Then

15
h ¡ RAq 0, for if h | RAq = 0, then by the Reduction
Theorem (Chapter 1, Theorem 1.12.) there would,', be an open
V 1 RA such that h I V* = 0. It would then follow
o 1
from Theorem 1.1. that R^-1^ V would be an open set
containing AQ. If R^”"^V is designated by U then
R U < V so that there is an A in 7 with A < U and
RA C RU C V*; therefore h [ RA = 0, a contradiction.
Case 1. Card Aq = 1, i.e., Aq = { x } . By
(ii) H^(RA ) = 0, a contradiction.
Case 2. Card Aq > 1. Write Aq = A^ W A0
where both A^ and A2 are proper closed subsets of A .
We consider the following part of the Mayer-Vietoris exact
sequence (Chapter I, Theorem 1.10.),
h°(ra1 n ra2) A, h1(rao) _£* h1(ra1) x h1(ra2).
Since by (iii) and (iv)
RA^ RAg = IJ | RA a Rb | (a, b) £ A^ x Ag | is
connected, then A = 0 (Theorem 1.11.), and
h RA e Ker J = ImA = 0,
o
a contradiction.
RX is a continuum and H^RX) = 0 imply that
RX is unicoherent (Theorem 1.15*)• RX being of codimension
one and H^(RX) = 0 imply that H^K) = 0 for every closed
subset K of RX [21], and thus every subcontinuum of RX
is unicoherent.

16
We now prove that every two points of RX are
separated in RX by a third point. Suppose there were
two points a and b such that no point separated a
and b in RX. Then by Lemma 2.3-, for any p different
from both a and b, there would be a continuum P which
would be irreducible from a to b and which would not
contain p. If q were an element of P distinct from
a and b there would also be a continuum Q, irreducible
from a to b and which would not contain q. But then
P Q would be a subcontinuum of RX which would not be
unicoherent, since P n Q by our selection of P and Q
is obviously not connected. This contradiction completes
the proof.
Theorem 2.4. If X is a compact Hausdorff
space, and if P is a relation on X, then the following
conditions
(i)X is of 1-codimension and s.l.c.,
(ii)P is a closed partial order,
(iii)P is left monotone, i.e., Px is connected,
and H^(Px) = 0 for every x in X,
(iv){Px | x e X) has the f.i.p., and
(v)P is right monotone, i.e., xP is
connected for every x in X,
are necessary and sufficient conditions that X be a

17
p
tree, and that P be a cutpolnt-order.
r •
Proof: We first prove the sufficiency.
Conditions (ii), (iv) and the first half of (iii) imply
that
Pa Pb = U { Px ( x e Pa n Pb }
is connected, and thus Lemma 2.2. yields that X is a
tree.
Since X is compact and { Px I x £ X} has the
f.i.p., then
{ Px | x e x } £ □
Indeed, it is a single point, the unique P-minimal element
of X. Let us denote by 0 the set
{ Px I x e x } .
We prove that P = Q(0) . If (a, b) £ Q(o) such that
a = 0 or a = b, then clearly (a, b) must be also in P.
If a separates 0 and b in X, then since Pb is a
continuum containing 0 and b, it must contain a, and
we again conclude that (a, b) is in P. Thus Q(0) C P.
Conversely, if (a, b) is in P, then since a is in
aP r\ Pb, and since both aP and Pb are continua, then
aP ^ Pb is a subcontinuum of the tree X, and therefore
2. Condition (v) is not necessary for X to be
a tree.
used.
3. Only reflexivity and transitivity of P are

18
by Lemma 2.1. it is unicoherent. Thus aP r\ Pb is also a
continuum. Now, by virtue of the Hausdorff-Maximality-
Principle, aP r\ Pb has a maximal P-chain, C, and
(a) C is closed (Theorem 1.5«)> and
(b) C _is connected.
For if C were not connected there would be two non-void
disjoint closed sets A and B such that C = A U B
and b £ B. The set A contains a maximal element m.
Define A' and B; by the equations
A/ = Pm o C and B/ = C - Pm.
B ‘ C mP, and since A C. A1 , then B/ C B. Now
A1 r\ B1 C Pm n (mP r\ B) = (Pm rN mP) r\ B = Q ,
therefore
C = A' U b'
is a separation. If bQ designates the minimal element
in B', then by the maximality of C
mP r\ Pb = { m, b }
o ' o J
which contradicts the connectedness of mP r\ Pb .
o
Therefore, any maximal P-chain in aP r\ Pb is connected.
(c) aP r\ Pb has a unique maximal P-chain,
which we denote by Cp(a> b).
If C and C' were two distinct maximal P-chains
in aP n Pb, then both C and C/ would contain a and
b, and C U C' would then be connected, and hence C n C'
would be connected. But for x £ C - c'

19
C ^ c' = (Px r\ C C) u (xP r\ C C') =
C ^ (Px U xP) r\ C1
is obviously a separation, a contradiction.
Since (0, b) £ Q(0) C P and since X is a
tree, there is a unique connected Q-chain [22], [23]
Cq(0, b) C Pb which contains both 0 and b. Pb must
also have a connected P-chain containing both 0 and b
and this P-chain must be unique. We denote by C (0, b)
the unique connected P-chain in Pb containing 0 and b.
Since a Q-chain is also a P-chain, then
Cp( 0, b) = CQ(0, b) .
Similarly, there is a unique connected P-chain Cp(0, a)
in Pa containing both 0 and a. It is clear that
Cp(0, a) V Cp(a, b) = Cp(0, b) = CQ(0, b).
As a consequence, a £ Cq(0, b) and hence (a, b) £ Q
which was to be proved.
We next prove the necessity. Let X be a tree
and let P be the cutpoint order on X with respect to
a point z in X. We prove that X and P satisfy the
conditions (i), (ii), (iii), (iv) and (v) stated in the
theorem:
Proof of (i). By Ward [24] a tree is a compact
connected commutative idempotent semigroup with zero,
therefore it is acyclic [17]. Hence in particular

20
H^(X) =0. We now show H^A) = 0 for every A = A* C X
and thus X is of one codimension, unless Xis degenerate.
Suppose on the contrary that H (A) ^ 0 for some closed
subset A of X. If h is a non-zero member of H^(A),
then by the Floor Theorem (Theorem l.l6.) there is a floor
F C A for h, which is connected. The set F being a
subcontinuum of a tree is itself a tree and hence is acyclic.
Therefore, H^F) = 0 which contradicts the fact that F
is a floor, and thus H^A) = 0. The semilocal connectedness
of X follows from the fact that X is compact and locally
connected.
Proof of (ii). This is proved in Ward [24].
Proof of (iii). The outpoint order P is order
dense [23] , and since by (ii) P = P* we have that every
maximal P-chain in Px is connected [22], thus Px is
connected. Indeed, Px itself is a tree and therefore as
has been proved in (i), H^Px) = 0.
Proof of (iv). This is obvious, since P has
the least element z.
Proof of (v). Replacing Px by xP in the
argument of (iii), we easily obtain the connectivity of xP.
Proof of Lemma 2.3» (l) There is a_ component
Aq of A such that M does not separate Aq and B in
X. For if otherwise, to each component A^ of A there is
a pair of disjoint open sets and T^ such that

X - M = U and } A^, T^ ^ B. Since A is
either compact or consisting of finitely many .components,
there is a finite subfamily { G^, G^, . .., G \ of G-^s
21
such
that
A C
v {
Gi 1
i =
1, 2, ..
., m } .
Whence
G =
m
G. and
i=l 1
T =
m
r\
i=l
T.
1
are two
disjoint
open sets
such
that
X - M
= G
KJ t
and
G P A,
T p B
so that the
hypothesis (i) is contradicted.
Throughout the rest of the proof, let #[S] be
the number of components of the space S. Let If = { | A
be the collection of all open sets containing M but
missing A such that #[X - V^] is finite. For each
£ If let R^ be the component of X - that contains
the component Aq of A. Designate R ^ u{R^ A e A }
(2) R i_s open. Let y e R, and let £ If
be such that R^ contains y. By the normality of X
there is a V, e If with V C V* C V, . Since X - V,
[X |X |X A [X
has only a finite number of components and since
(x - y° 3 X - vj 3 x - Vv
we have R, C R° C R. Therefore R is open.
A ^ P
(3) For each VQ e If , there is an R^ containing
R such that
a
#[>V n (X - Va)] = n
22
for all containing R^. Let C-^, Cg, . .., Cn be the
n components of X - V , so that
X - Va = C-^ o Cg • • • U
where = R^, then for any contained in VQ we have
X - containing X - Va, so that each Cf must be either
totally contained in R^, the component of X - contain¬
ing Aq, or disjoint from R^. Thus
#[Rp rs (X - Va)] < # (X - Va) = n.
But,
# [r6 rv (x - va) ] < § [rt r\ (x - va) ]
if R^ 2 R5 ^ R . Hence there is an R^ } R^ such that
the number #[R^ r\ (X - VQ)] is the maximal so that
#[ for all R^ containing R^.
(4) For each VQ e there is an R^ such that
«n = Rx ^ (X - va)
for all R^ containing R^. By virtue of (3) above, there
is an R, containing R such' that
A Ct
#t>V ~ (X - va>] = #tRA ^ for all R containing R^. We note, further, that
*V ^ (X - Va> ^ (X - Va)
for all R^ containing R^. For if as in (3) above,
we express
X - Va = ci U c2 U ... U Cn
where, without loss of generality, CL is contained in R^

23
for i = 1, 2, . . ., k (k < n) and Ck r\ R^ = O for
i = k + 1, ..., n, then it follows from R .containing
[X
R-^ and
that
#[R^ ^ (X - VQ)] = # [Ra ^ (X - VQ)]
C. r\ R = Q for i = k + 1, . .., n. And therefore,
1
Rp. ^ u c
k
= RA kN (X - Va) .
(3) R = R* - M; that is, R _is closed in X - M.
If y is a point of R - M then there exists a Va in
V- missing y. By the normality of X there is a V,
P
in such that C V^* C VQ. By (4) there is an R^
containing R^ such that
A (X - Vp) = R* A (X - Vp)
for all R^ containing R^. Furthermore, if U is an
open set about y, then U ^ R^ 41 O • For, if we designate
the set U r\ (X - V^*) by W then y is in both R - M
and W so that W ^ R 4= O , and there is an R^ such
that W n R¿ 0 • Without loss of generality we may
assume that R, contains R, . Then
o A
□ * W n Ró = W ^ Ra C (X - Vp)
= W r\ R^ r\ (X - Vp)
= W A R .
Thus, U ri R^ } W A R^ 4: G , so that y is in R^ = R^.
Therefore, R is closed in X - M.

It is to be noted that R meets B and hence
R T r\ B ^ Q for some t ; because otherwise.
X - M = R U(X-M-R)
would be a separation, of X - M between Aq and B, so
that (l) would be contradicted. We now conclude
result by taking N = R^ .
the

CHAPTER III
FIXED POINTS AND MINIMAL ELEMENTS
In the Symposium of General Topology and its
Relations to Modern Analysis and Algebra (Prague 196l),
Professor A. D. Wallace announced [18] among other things
the following fixed point theorem.
Theorem 3-1» [18, Theorem 5]* If X is a
continuum, if P
is a closed
left
monotone partial order
on X such that
PA* C (PA)*
for
each
A C X, and if
z separates Pa
and Pb in
X,
then
Pz = z.
Wallace applied this theorem to prove
Theorem 3 - 2 . [18, Theorem 6]. If X is a con¬
tinuum and if P is a closed left monotone partial order
on A such that PA*C (PA)* for each A ( X, then the set
K of P-minimal elements is connected.
However, in [18] Wallace gave no proof of the
fixed point theorem. For the sake of completeness, we will
give in this chapter a proof of Theorem 3*1 by first proving
Theorem 3-2 and using this result to establish Theorem 3•1•
We will also show that under certain conditions the state¬
ments
(i) If z separates Pa and Pb in X then
25

26
Pz = z.
r »
(ii) The set of P-minimal elements 'is connected,
are equivalent (Theorem 3**0-
Furthermore, if P is a partial order on the
continuum X and K is the P-minimal elements in X,
we will prove in Theorems 3-5^ 3*6 and 3-7 that the
cohomology groups of X and those of K are isomorphic
for all non-negative dimensions for certain classes of X
with suitable choice of P.
The following lemma will be used in the proofs of
foregoing theorems.
Lemma 3•3• [cf. l6]. If X is a compact Haus-
dorff space and if P is a. lower semi-closed partial order
on X such that PA* < (PA)* for each A ( X, then the set
K of P-minimal elements is closed.'
Proof. Suppose by way of contradiction that there
is an x in K* - K. Then since Px is closed and X is
compact, Px has a minimal element which must be a P-minimal
element in X, and thus
Px r\ K if Q .
Let y be any element in PxnK. Obviously x ^ y, so
that there is an open set U containing x whose closure
excludes y. If V = U r\ K, then PV = V, and x is in
V*.
Since

27
PV*C (pvf = V*,
then PV* C U* and hence y e Px C PV* C U*". This contra¬
dicts the fact that y ^ U*. Therefore the set K is
closed.
Proof of Theorem 3*2. Let us assume contrary to
the conclusion of the theorem that K is not connected.
By Lemma 3-3 there are two disjoint non-void closed sets
A and B such that K = A ^ B. Since P is closed, then
AP and BP are closed and X = AP w BP. Furthermore, the
connectedness of X yields AP os BP % â–¡ . Thus AP rs BP
has a minimal element, say t. Since
Pt = (Pt r\ AP) v (Pt BP) ,
then
(Pt AP) rs (Pt BP) = Pt rs (AP rs BP) = t.
If we designate
C = (Pt r\ AP) - t and D = (Pt/^ BP) - t,
then both C and D are open in Pt. Furthermore, C and
D are both non-void. For if C is void, then Pt r\ AP = t,
and so t must be in A. This implies that t is not in
BP which contradicts the fact that t is a minimal element
in AP r\ BP. Similarly, D is not void. Consequently,
Pt - t = C U D is a separation. The connectedness of Pt
yields C* = Pt r\ AP and D* = Pt r\ BP. Now, for each x
in Pt - t we have Px C Pt - t and the connectedness of
Px then implies that either Px C C or Px C D. Thus,

28
for each x in C we see that Px C C, and hence PC C C,
and it follows that
C* 3 (PC)* b PC * = P(C ^ t) ) Pt 5 D
which is a contradiction. Therefore K is connected.
Proof of Theorem 3-1- Let X - z = A ^ B where
A and B are non-void separated sets and aP ( A and
bP C B. Let K be the set of P-minimal elements. Now,
if Pz ^ z, then K C X - z and
K = (K r\ A) V (Kn B)
is a separation for K which contradicts the connectedness
of K.
Theorem 3•4. If X is a continuum and P is a
closed left monotone partial order on X such that the set
of P-minimal elements K is closed, then the following
statements are equivalent:
(i) If z separates Pa and Pb in X then
Pz = z.
(ii) The set K is connected.
Proof. (i) implies (ii). Suppose the set K is
the union of two disjoint non-void closed sets A and B.
Since K is the set of P-minimal elements then
X = KP = (A ^ B)P = AP U BP.
And the hypotheses that X is connected and P is closed
yield AP r\ BP 4= □ • Let z be a P-minimal element of
AP BP. Then there are elements a e a and b^B such

29
that {a, b}< Pz - z. The equalities
Pz = (Pz ^ AP) b (Pz n BP) , and
Pz a (AP r\ BP) = z
show that
Pz - z = (Pz r\ AP - z) ^ (Pz A AP - z)
is a separation, that is, . z separates a = Pa and b = Pb
in the continuum Pz, and so by (i) z is a minimal element
which is neither in A nor B. This contradiction estab¬
lishes the connected ness of K.
(ii) implies (i). The proof as given for Theorem
3-1 applies here as well.
We now present an example to motivate Theorem 3*3.
Example. Let X = { (a, b) 1 a2 4- b2 = 1 ^ U
{(0, b) | 1 <_ b <_ 2} for reals a and b. Let X be
endowed with the Euclidean topology. Let
P = { ((0, b1), (0, b2)) | 1 < bx < b2 < 2j U A
where A = | (x,x) | x £ X } . Then P = P* is a both left
and right monotone partial order on X such that PA C (PA)
for each A ( X and
K = { (a, b) l a2 + b2 = l} = S1 (l-sphere)
is the set of P-minimal elements.
It is to be noted in this example that H^Px) = 0
for each x e X while H^(K) = H^(S) + 0 for any non¬
trivial coefficient group. However the following equality

30
Hp(x) = HP(K) holds for all non-negative dimensions. .
It is interesting to seek conditions on X and
a relation P which imply the equality HP(X) - HP(K) for
all non-negative dimensions. The purpose of the next
theorem is to take a small step in this direction.
Theorem 3-5- If X is a continuum, and if P
is an upper semiclosed partial order on X such that
(i)PA* C (PA)* for all A C X,
(ii)there exists an element u in X such that
X - K C Pu where K = {x| Px = x }, and
(iii)the quotient space X/K x K u A modulo
K X K u A is a topological semilattice under the natural
partial order induced by P, where A - { (x, x) I x 6 X } ,
then, Hp(x) s H^(K) for all integers p > 0.
Proof. For simplicity in notation we write
Y = X/KXKUA. Since every compact topological semilattice
has a (unique) zero, we write z for the zero of the semi¬
lattice Y. By the hypothesis (ii), Y has a (unique) unit
which will be denoted by u also since no confusion is
likely to occur.
We will accomplish the proof in three steps.
(a) H^(X, K) = HP(Y, z) for all integers p >_ 0.
Let f : (X, K) -> (Y, z) be the natural map. Since (X, K)
is a compact pair, f is a closed map, and moreover, f

31
takes X - K topologically onto Y - z. Thus-by the Map
» *
Excision Theorem (Theorem 1.13) f*- : HP(Y, z) -> HP(X, K)
is an isomorphism.
(b) HP(X, K) = 0 for all integers p > 0.
We
establish this by showing HP(Y, z) = 0 for all integers
p >_ 0. Define ffc : (Y,z) -» (Y,z) by f (y) = t a y for
all t e Y, where a is the semilattice operation on Y.
If for each t £ Y define A^: (Y, z) -> (YX Y, z xY) by
A^(y) = (y, t) , then f = A o A^. Since by the Homotopy
Lemma (Theorem 1.14) Az*= Au*, thus
f *= A*. A*= A*. A*» f *.
z z u u
Since f^ is the identity map, so is f z. Let
i : (z,z) C (Y, z) be an inclusion map and let h : (Y, z)
(z, z) be defined by h(y) = z for all y e Y. Since
f = i • h, the following diagram
áj
is analytic, for all non-negative integers p, that is
fj = f *= h* * i* . Since Hp(z, z) = 0 for all integers
p > 0 and HP(Y, z) = fJ(Hp(Y, z))Ch*(Hp(z, z)) for all

32
integers p >_ 0, then HP(Y, z) = 0 for all integers
p >_ 0 as desired.
(c) HP(X) = H^K) for all integers p >_ 0. By
combining the above results with the exact sequence for
the triple (X, K, â–¡ ) (Theorem 1.7)* we obtain the exact
sequence
0 = HP(X, K) -» HP(x) •> HP(K) -> HP+ 1(X, K) = 0
for p = 0, 1, 2, ... . Consequently, Hp(x) and HP(K)
are isomorphic for all non-negative dimensions.
We now center our attention on the case in which
X is of codimension one.
Theorem 3«6. If P is a closed partial order on
a continuum X and if ir : X^X/KXKUA designates the
natural map, such that
(i)PA*C (PA)* for all A C X,
(ii)P is left monotone and H^(7r(Px)) = 0 for
all x e x,
(iii)X is of codimension one,
then Hp(x) « HP(K) for all p > 0.
Proof. Denote X =X/KXK^A and
P = { (tr(x), ir(y)) J (x, y) e p J
then since X is compact and ir continuous, P is closed
and left monotone; indeed we have P7r(x) = tt(Px) . By
virtue of Theorem 1.17* we have HP(X, K) = HP(X)
for all

33
p >_ l. Since X is of codimension one we hav.é HP(X, K) =
0 for p >_ 2 [21] . We show H^X, K) = 0 by proving
H^(X) = 0 . Using the same argument employed in the proof
of Lemma 2.2 one sees that H^(PS) = 0 for all closed sets
"1 ^
S in X; in particular we have H (X) = 0 . We have noted
in part (c) of the proof for Theorem 3*5 that if HP(X, K)
= 0 for all p > 1 then HP(X) s HP(K) for all p > 1.
The equality H^(x) = H°(K) follows from the fact that
both X and K are connected [Theorem 3*2].
We remark that the hypothesis (iii) in Theorem
3-6 may be weakened to :
(iii ) cd (X - K)* = 1,
and this may even be dropped completely if each Px is a
chain, as will be seen in the following
Theorem 3 «7• If P is a closed partial order on
a continuum X such that
(i) PA* C (PA)*, and
(ii) each Px is a connected chain,
then Hp(x) £ HP(K) for all p > 0.
Proof. Let X, 7r and P be defined as in
Theorem 3-6. Since P is closed and 7rPx is a connected
chain P is closed and Ptt(x) is a connected chain. It
follows then that each P7r(x) is a generalized arc and
hence Hp(P7t(x)) = 0 for all p >_ 1. As it has been noted

3^
in the proof of Theorem 3*6, H1(PS) = 0 for ¿11 closed
sets S in X. We now show H^X) = 0 for all p >_ 1
by proving HP(PS) = 0 for all p >_ 1 and for each closed
set S in X. If there were a least integer n such that
Hn(PA) i= 0 for some closed set A in X, then n would
have to be greater than 1, and A ± □ . Let h be a non¬
zero member of Hn(PA), then h | Pa = 0 for each a £ A
and hence by the Reduction Theorem there is an open set V
containing Pa such that h I V =0; then by Theorem 1.1
there is an open set U containing a such that PU C V.
Thus the collection a of all open subsets U of A such
that h J PU = 0 forms an open cover of A. Also Q. is
closed under finite union, for if and Ug are in a
denote hQ = h | PU* U PU*. Since P^ U Ug)* = P(U* ^ U*)
te PU-^ ^ PUg , to show G is closed under finite union it
suffices to show hQ = 0. In the following part of Mayer-
Vietoris exact sequence :
Hn-1(PU*^ PU*) *» Hn(PU*^PU*) í Hn(PU*) x Hn(PUg)
since J*(hQ) = (h I PU*, h I PUg) = (0, 0) , hQ is in the
image of A. But Hn_1(PU*^ PU*) = Hn_1(PS) where S =
PU^r\?uJf is a closed subset of X, and hence Hn-'L(PS) =
0 by the minimality of n. This proves hQ = 0 so that
a is closed under finite union. Since A is compact,
A is a union of some finitely many elements of (X • Thus
A must be in G and so h = 0, a contradiction. This

35
together with Theorem 1.17 implies HP(X, K) = Hp(x) = 0. for
all p >_ 1. It then follows from the exact sequence for the
triple (X, K, O) [Theorem 1.7] and the connectedness of X
and K [Theorem 3-2] that Hp(x) = HP(K) for all p > 0.
Remark. Theorem 3-7 may be stated more generally
by repacing the hypothesis (ii) by
(ii') each 7r(Px) is acyclic.
We conclude this chapter by exhibiting an example
which answers a question in Topological Semigroups. A clan
is a compact connected topological semigroup with unit [20].
Let S be a clan and let
R = { (x, y) | (x, y) e S X S, xS uSx c yS u Sy }
then R is a closed quasi-order on S and the set of
R-minimal elements is also the minimal ideal of S. The
question to be answered is: if. a. clan S is. a. tree, is
its minimal ideal an arc or a. point 9
The answer is affirmative if S is abelian (or
normal: xS = Sx for all x e S), but it is negative in
general.
Example. Let S = X ^ I be the subset of
Euclidean 3-space such that
X = { (x,0,0) | -1 < x < 1 } U {(0,y,0) | -1 < y < 1}
I = { (0,0,t) | 0 < t < 1 } ;

36
let S be endowed with the
Euclidean topology and let
the multiplication o on
S be defined below (the
usual multiplication of
reals is denoted by
juxtaposition) :
i.Aop =
ii.(I, o)
unit interval; i.e.,
A for all A £ X and for all p e S
is the usual semigroup of the real
(0,0,tx)o(0;0,t2) = (0,0,t1t2),
iii.For each (0,0,t) e I and for any (0,y,0)ex,
(0,0,t)o(x,0,0) = (tx,0,0) and (0,0,t)o(0,y,0) = (0,ty,0).
Then, (l) (S, o) is associative.
(2) o : S S -> S is continuous.
(3) (S, o) is a clan with (0,0,l) as unit.
(4) The minimal ideal of (S, o) is X which
is neither an arc nor a point.
It is interesting to observe that the semigroup
S given in the above example may be realized as a semi¬
group of matrices by the following one-to-one correspond¬
ence :
■»
t
x + y/^1
(x,y,t) +
0
1

37
where -l<_x, y <_ 1, 0 < t < 1 and (tx)2 + (:xy)2 + (yt)2
= 0. The correspondence is indeed an isomorphism. Prom
this the assertions (l), (2) and (3) in the example
are self-evident.
/

CHAPTER IV
FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES
A space X is said to have thé _f. jd. jo. (fixed
point property) if, for every continuous function f: X -> X
there exists some x in X such that x = f(x). Hamilton
[8] has proved that the chainable metric continua have the
f. p . p .
A space X is said to have the F. jo. jo. (fixed
point property for multifunctions) if every continuous
’multifunction'*' F : X -> X has a fixed point, i.e., there
exists a point x in S such that x £ F(x). Obviously
if X has the F. p. p. then it has the f. p. p., but
the converse need not be true. Strother [13] has exhibited
two continua X and
Y
both have the
F.
P-
P-
but their
Cartesian product X
x Y
fails to have
the
F.
P-
p. Borsuk
[3] has constructed a decreasing sequence of three-cells
whose intersection does not have the f. p. p. The inter¬
section is the inverse limit, the bonding maps being in¬
jections. As a counter theorem to the results of Strother
and of Borsuk, we prove that if (X^, tt-^, /\ ) is an
1. Following Strother [12], a multifunction
F : X -> Y is continuous if, and only if, F(x) is closed for
each x, and F "*'(A) is open (closed) if A is open (closed).
38

39
inverse system of compact spaces such that each has
the F. p. p. then the inverse limit space has the F. p. p.
As a corollary to this, we obtain Ward's generalization [25]
of the Hamilton theorem [8] that every chainable metric con¬
tinuum has the F. p. p. Our result is, indeed, stronger
than the Ward's, since it includes some of the non-metric
chainable continua as well.
Definition 4.1. The collection (X-^, tt^ , A- )
is as inverse system of spaces if:
(i)A is a directed set,
(ii)A in A implies that X^ is a Hausdorff
space,
(iii)whenever A > p there is a continuous
function 7T-y : X, -> X, ,
Ap A [1
(iv)if A > p and p > v, then tt^ = •
The function 7r^ is called a bonding map. If A
is in A let be the subset of the Cartesian product
P{ X^ | A e A } , defined by
= | x j if A > p then tt-^x(A) = x(p)} ,
where x(A) denotes the A-th coordinate of x.
Definition 4.2. The inverse' limit space X^ of
the inverse system of spaces (X-^, tt^ , A ) is defined to
be
x„ = r\ { sA i * 6 a }

40
endowed with the relative topology inherited fpom the
product topology for P { | A e A } ; in notation X
= Iim (X^, ir^ , A ) .
We write p^ : P { X^ | A € A } ■* X^ for the A-th
projection of P [ X^ | A e A } , i.e., p^(x) = x(4) for aH
xin P|X^| A^A}; the restriction p^ j X^ will be de¬
noted by 7which will be called a projection map. It is
readily seen from the definition that an element x of
P { XA | A e A] is in X^ if and only if = tt (x)
whenever A > p. A more detailed account of inverse limit
space may be found in Lefschetz [10], Eilenberg and Steenrod
[7], Capel [4] and Mardesic [11].
The following known results (see, e.g., [4], [10])
will be used.
Lemma 4.3» (i) The collection ^ir^(U^) ¡ A £ A
and is an open subset of X^ j forms a basis for the
topology of X».
(ii)The inverse limit space X^, is Hausdorff;
if A € A , is a closed subset of P { X^ [ A e A} so
that X^ is closed in P •{ X^ | A e A } •
(iii)If X^ is compact for each A in A then
X^ is compact; if, in addition, each X^ is non-void
then Xjo is non-void.
(iv)If X^ is a continuum for each A o A then
the inverse limit space is a continuum.

Lemma 4.4. If A is a compact subset of X^
and if t| , then (tt^(A), ir^' , A) is an
inverse system of spaces such that A = 11m (ir^(A), tt^, A)
and each bonding map ir^ is onto.
In the sequel, since we are only interested in
compact spaces, each projection map tt-^ will be assumed
to_ be_ onto; for if otherwise, by virtue of Lemma 4.4,
each X^ may be replaced by tt^(X*.) without disturbing
the resulting inverse limit space. We are now ready to
state our main result.
Theorem 4.5» Let (X^, , A) be an inverse
system of compact spaces such that each X-^ has the P. p.
p., then the inverse limit space X*, also has the F. p. p.
We divide the proof of this theorem into the
following steps. In Lemmas 4.6, 4.7 and 4.8 will be
assumed to be the inverse limit space of the inverse system
(X^, tt-^, A ) of compact spaces.
Lemma 4.6. If F : X^ -> X^ is a continuous
multifunction, define F^ : X^ -> X^ by F^ = v-^F v^ for
each A. Then F, is a continuous multifunction.
A
Proof. (i) If t is in X^, then since F : X^ -*• X„
is a continuous multifunction, by Theorem 1.1 and Theorem
1.4 of Chapter I, Fir~^(t) is a closed subset of X^. Thus,

F4tJ
is closed for every
t in
X, .
(ii) If CA is a
closed
subset of XA, then
is closed. It is
readily
seen that F ^tta^(Ca)
Is closed in Xm and hence compact; therefore
•7r^F~'Sr^ (C^) = F^(C^) is compact and hence closed.
(iii) If UA is open in XA, then F^(U^) is open.
7rA and F being continuous, F ■'‘Tr-^(U^) is an open subset
of Xoo • It follows then, by virtue of Lemma 4.3 (i), that
ir^F-1TT^1(U^) = F^(U^) is open.
Thus, by (i), (ii) and (iii) above, F^: X-^X-^
is continuous.
Lemma 4.7» Let F : Xffl -> X«, be a continuous
multifunction, let F^ : X^ -> X^ be defined as in Lemma 4.6
Then, for each x in Xa, ,
(i) (F^rr^(x), ttA[x, A)2 and (ir^F(x) , v^, A )
are inverse systems of compact spaces,
(ii) lim (FA7rA(x), 7tA[x, A) = 11m (ttaF(x) , tta , A)
(iii) F(x) = lim (Fa7Ta(x), 7Ta^, A).
Proof. Since each FA is continuous (Lemma 4.6)
and each XA is compact, so is F-^ir-^(x) for all A e A .
2. For simplicity in symbolism, henceforth if
A C lim (XA, 7r A^, A) then (u^A, tta , A) will mean
KA> a ) •

43
To show that (F^tt-^(x) , 7r^(X> A ) forms ah inverse system,
it suffices to show 7T^^F^tt^(x) £ F^7r^(x) whenever A > |x.
To this end we first observe
Vx> < K^aiAaM - n£Vx)’
sincé 7T-, 7T-> — ir • From this we have
Ap A [X
’’aixVaW c ’rAnVÁÍ’V(x)
= lrA1>A1' ’rÁ1),rÁÍ’rM.(x)
= Ati,rA)P(lrAM.’rAr\(x>
= v ’vSiW
= F 7T (x) ,
(X |X
by the definition of FA, F^ and the equality = 7r^.
The fact that (7T-^F(x) , tta , A) forms an inverse
system follows from Lemma 4.4.
(ii) For each A e A and any x e X^, , we have
x e tta tta(x) and hence,
7T.F(x) C u\F irt^Ax) = (tt,F ir~±)vy{x) = F,ttAx)
-1>
Thus,
lim (tt^F(x), tt^, A)C^ (Fatta(x), tt^, A).
To prove the other inclusion, we show
Xoo - lim (ttaF(x), ttA[i, A) C Xm- lim (F^r^(x) , ir^, A).
Let y be in Xa - lim (ttaF(x), , /\ ) then, by Lemma 4.4

44
there exists a p e A such that ir^(y) ^ tt^FÍx) . Let U|
M-
M-
and V be two disjoint open sets in X such that
P M-
so that
Vy) 6 um. and V(x) c V
F c •
It follows then from Lemma 4.3 (i) and the continuity of
F that there exists a 6 £ A and an open set in X^
such that x a tt^U^ , and
<*) F(u¡\) c ir'\ •
Since A is directed, there is a Aq e A such that
Aq > p and Aq > 6, we shall use this Aq throughout the
proof of this lemma. If we write U.. = ir-T^eU* and use
Aq AqÓ 0
the equality ttT1 = tt-.1 irr,1,- , then ( Hr) may be rewritten
0 aq Aq0
as
F<< V c r-\ ,
-1,
and hence
F, U, = 7T, F 7T,1(U, ) ( 7f, 7T 1V = 7T, (iT, TT, ) 1V
Aq Aq Ao A0 Aq V Aq P P A0 AqP Aq p
- •
In particular,
F, TT, (x) ( IT,1 V .
Ao Ao ^ Ao^ ^
Similarly, one obtains
^o(y) 6 T^un ’
since ^(y) e D(i and ^ = v ^
The fact that U
are disjoint implies TTA0pVpn^AopUp
—.UM = □ and
and

45
consequently 7r, (y) ¿ F, tt, (x) . From this we conclude
Ao * Ao Ao
y ^ lim (F^tt^(x), tt^ , A ) as desired.
(iii) This follows immediately from (ii) and
Lemma 4.4.
Lemma 4.8. Let F : X^-* X,*, be a continuous multi¬
function, let F^ : X, -> X^ be defined as in Lemma 4.6 Let
Ex = íex l ex e xx and ex € Vex>} fchen
forms an inverse system.
Proof. It suffices to prove £ E^ whenever
A > p. Let e^ £ F-^(e^), then
V^x) 6 TxnFx c : VVr*1(e*)
= WaK1*^ = V'xn^xh-
Thus, ir^E^ C E|j_ as is to be proven,
Proof of Theorem 4.5» Since each X^ has the
F. p. p. and by Lemma 4.6 each F^ : X^ -» X-^ is continu¬
ous, each E^ is closed and non-void. By Lemma 4.8,
(E^, , A ) is an inverse system of compact spaces, so
it has a non-void inverse limit space lim (E-^, tt^, A ) .
We now conclude the proof by showing that each x in
iim (Ea, ttA/U, A ) is a fixed point under F; i.e., x£F(x).

46
If x is in lim (E^, tt^, A) then tta(x) for-all
A G A ; i.e., tta(x) e E^7r7v(x) f°r &H A G A . Conse¬
quently, by Lemmas 4.4 and 4.7, we have
x = lim (irA(x), tt^, A) g lim (F^tt^x) , vA)
= F(x).
In fact, with the assumption of Theorem 4*5 and
the notation of Lemma 4.8 together with the notation
E = | x j x G F(x) J. , we can make the following sharper
assertion.
Theorem 4.5' . E = lim (E^, tt^, A ) .
Proof. Prom the proof of Theorem 4.5, we have
E } lim (Ea, ttA[X, A) .
It remains to prove that
E C lim (Ea, tt^, A ) •
Let x be in E, then x G F(x) and therefore, for all
A e A ,
^(x) g tt^F(x) C = Fa(tta(x)).
That is, tta(x) g E^ for all A; consequently, by Lemma 4.4
E C lim (Ea, ir^ , A ) •
A chain (U^, U^, •••, U ) is a finite sequence
of sets U. such that U. r\ U. ± □ if and only if
J
| i - j [ <_ 1. A Hausdorff space X is said to be chainable

47
.if to each open cover \J* of X there is a finite open
cover /[X = (U^, Ug, . .., Un) such that (i) U refines
V"> (ii) u= ( Uf, Ug, . .., Un) forms a chain. It
follows that a chainable space is a continuum. It is
implicit in a paper by Isbell [9] that each metrizable
chainable continuum is the inverse limit space of a sequence
of (real) arcs. This together with a theorem of Strother
[13] that a bounded closed interval of the real numbers has
the F. p. p. imply the following result of Ward [23] as a
consequence of Theorem 4.5-
Corollary 4.9. Each chainable metric continuum
has the-F. p. p.
Examples of inverse limit spaces of inverse
systems of real arcs exist which are not metrizable; for
instance, the "long line" is one such. Thus, Theorem 4.5
is a proper generalization of that of Ward's [25]*

BIBLIOGRAPHY
1. Anderson, L. W. and Ward, L. E., Jr. One-dimensional
topological semilattices. Ill. J. Math., 5
(1961), 182-186. ,
2.
Borsuk, K. líber die Abbildungen
Raume auf die Kreislinie.
der metrischen kompakten
Fund. Math., 20 (1933),
224-231.
3. Sur un continu acyclique qui se laisse trans¬
former topologiquement en lui meme sans points
invariants. Fund. Math., 24 (1935), 51-58.
4. Capel, C. E. Inverse limit spaces. Duke Math. J., 21
(1954), 233-245.
5. Cech, E. Sur les continus Peaniens unicoherents. Fund.
Math., 20 (1933), 232-243-
6. Cohen, H. A cohomological definition of dimension for
locally compact Hausdorff. spaces . Duke Math. J. ,
21 (1954), 209-224.
7* Eilenberg, S. and Steenrod, N. Foundations of algebraic
topology. Princeton University Press, Princeton
1952.
8. Hamilton, 0. H. A fixed point theorem for pseudo-arcs
and certain other metric continua. Proc. Amer.
Math. Soc., 2 (l95l), 173-174.

9.
10.
11.
/
12.
14.
Isbell, J. R. Embeddings of inverse limits 4 Ann. of
Math., 70 (1959), 73-84.
Lefschetz, S, Algebraic topology. Amer. Math. Soc.
Colloq. Publ., no. 27, New York 1942.
Mardesic, S. On inverse limits of compact spaces.
Glasnik Mat. Fiz. Astr., 13 (1958), 249-255-
Strother, W. L. Continuity for multi-valued functions
and some applications to topology. Doctoral
Dissertation, Tulane University 1952.
On an open question concerning fixed points.
Proc. Amer. Math. Soc., 4 (1953), 988-993-
Wallace, A. D. A fixed point theorem. Bull. Amer. Math
Soc., 51 (1945), 613-616.
15- Cohomology, dimension and mobs. Summa
Brasil. Math., 3 (1953), 43-55-
l6. Struct ideals. Proc. Amer. Math. Soc., 6
(1955), 634-638.
17• Acyclicity of compact connected semigroups.
Fund. Math., 50 (l96l), 99-105-
18. Relations on topological spaces. Proc. Symp
on General Topology and its Relations to Modern
Analysis and Algebra. Prague 1961, 356-360.
19. Relation-theory, Lecture Notes. University
of Florida, 1963-1964.
20. Topological semigroups, Lecture Notes.
University of Florida, 1964-1965-

50
21. Wallace, A. D. Algebraic topology, Lecture Notes.
University of Florida, 1964-65»
22. Ward, L. E. Jr. Partially ordered topological spaces.
Proc. Amer. Math. Soc., 5 (1954), l44-l6l.
23- A. note on dendrites and trees. Proc. Amer.
Math. Soc., 5 (1954), 992-994.
24. Mobs, trees, fixed points. Proc. Amer.
Math. Soc., 8 (1957), 798-804.
25- A_ fixed point theorem. Amer. Math. Monthly,
65 (1958), 271-272.
26. Whyburn, G. T. Analytic topology. Amer. Math. Soc.,
1942.

BIOGRAPHICAL SKETCH
Shwu-yeng Tzeng Lin was born May 11, 193^ at
Tainan, Formosa. In June, 1953, she was graduated from
Tainan Girl's High School. In June, 1958> she received
the degree of Bachelor of Sciences from Taiwan Normal
University. She worked as a Research Assistant at the
Mathematics Institute of Academia Sinica from 1958 to
I960. In the fall of I960 she enrolled, and worked as a
teaching assistant, in the Department of Mathematics,
Graduate School of the Tulane University, New Orleans,
Louisiana, and received the degree of Master of Sciences
in May, 1963- From September, 1963, until the present
time she has pursued her work toward the degree of Doctor
of Philosophy at the University of Florida.
Shwu-yeng Tzeng Lin is married to You-Feng Lin
and is the mother of one child.
51

This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Arts and
Sciences and to the Graduate Council, and was approved as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
August, 1965
Dean, College of Arts
and Sciences
Supervisory Committee:
ú j\
Chairman
SLL. k-w/. c/ (-j/c-i óü.
Dean, Graduate School
&
*3, '

UNIVERSITY OF FLORIDA
V IT 11,1111mi mi
3 1262 08556 7369



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