Relations on spaces


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Relations on spaces
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iii, 51 leaves : illus. ; 28 cm.
Lin, Shwu-Yeng Tzeng, 1934-
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Mathematics thesis Ph. D
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Thesis - University of Florida.
Bibliography: leaves 48-50.
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Full Text





August, 1965


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.



ACKIOWLEDGMENTS ..................................... ii


I. PRELIMINARIES............................. 1


ORDER ON A TREE ......................... 11



LIMIT SPACES ................ ......... 38

BIBLIOGRAPHY .................................... 48

BIOGRAPHICAL SKETCH .................................. 51




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 ((A x X) (-\ R)


RA = p1 ((X x A) {\ R)

where p1, p2 are respectively the first

projections. For x in X we shall write

i x R.

and the second

simply xR for


R(-I) = { (y, x) I (x, y) E R ) ,

AR[~1] = { x xR C A .,
R1-A { y I 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

lower semi-continuous (abbreviated l.s.c.)

said to be upper semi-continuous (abbreviated u.s.c.)


at x in X if and only if x in { implies x in
(VR(-l)) o w -] ,,
S( O for all V = V C X. R is said to be

l.s.c. l.s.c.
{ } on X if and only if R is lsc at
u.s.c. u.s.c.
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 ( f and only if xR
Slower semiclosed (
is closed for any x in X.

Definition 1.3. A relation R on X is said to
be a partial orderI if and only if the following conditions
are satisfied:
(a) (reflexivity): (x, x) E R for all x in X,
(b) (antisymmetry): (x, y) E R and (y, x) E R
imply x = y,
(c) transitivityy): (x, y) E R and (y, z) E R
imply (x, z) e R.

1. A relation R is said to be a quasi-order if
(a) and (c) are satisfied.

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-I-1 V and VR[-1] are open.

Theorem 1.2. The following statements are equi-
(a) R is l.s.c. on X.
(b) RA C (RA) for all A C X.

(c) A* R C (AR)* for all A C X.
(d) RV is open for all open V C X.
(e) p, I R is open, where p, I R is the
restriction of p, to R.

Theorem 1.3. The following statements are equi-
(a) R is u.s.c.

(b) VR[-1] is open for all V C X.
(c) AR(-l) is closed for all closed A C X.
(d) A R[-1] ( (AR[-l])o 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,

then R = R* if and only if

(a) R is u.s.c. on X, and

(b) R is 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 and

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 (

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
be 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 ( A C X, then

H(X,A) -4 HO(X,B) H(A,B)6

-- - ------- -
HI1XA) j- H(XB) (A,IB) -


H n( XA) -------- ---
Hn(X,A) i
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) -? H(A) 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 a 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

i:(X, A) C (X, B).

Theorem 1.9. If f:(X, A,
continuous, and if u: (X, A) (X',
v: (X, B) (X',
w: (A, B) -> (A',

B) (X' A', B') is

are defined by
u(x) = v(x) = w(x) = f(x),
then the ladder

SHP(X, A) HP(X, B) HP(A, B) ---_

u* J V* W*
--,HP(X, A)- HP(X, B) -i HP(A, B) --

is analytic, that is, each rectangle of the ladder is

We state here special case of The Mayer-Vietoris
Sequence which will be sufficient in what follows for our

Theorem 1.10. If X is a compact Hausdorff space,
and if X = X1 U X2 where X1 and X2 are closed subsets
of X, then there exists an exact sequence

I-- )Hq-l(x 1-, X2) H q(X) J-T Hq(x1) X Hq(X2) --

Hq(X, 1 xn ) A

where (j i 6) and (j 6) are the homo-
morphisms in the exact sequences for the triples (X, Xa, j )
and (X X1 r X2, 0 ) respectively for a = 1i, 2;
k: (X2, X1 r X2) C (X, X1) ; J*= j*l x j*2
I* i* i k -1-
= i- 1 ; and A = J k 62

As a consequence of the foregoing theorem we have
the following

Theorem 1.11. With the hypotheses of Theorem
1.10 and if X1 \ X2 is connected, the homomorphism A in

H(X1 r X2) H H (X(x ) X H (X2)

is 0.
Proof. Observe the following ladder

H(X) -- Hl(X, X1) -1 HI1() 1)

62 H -<> -- T
H(x x2) 2 X, XI X) Hx)

where u: X1 r\ X2 C X and where (j*, i, 6) (J 2' 2)
and k are as in Theorem 1.10. It follows from Theorem 1.8
that if h is in H(X r\ X2) then there is an h' in
H(X) such that u*(h') = h. Thus by Theorem 1.9., we have

k*- 62(h) = k k 6(h

which contains 61(h'). And A = j k* -1 62 being

a well-defined homomorphism, we see that for any a and P

in k*-' 32(h), J*(a) = J(P). Therefore,

A(h) =j k 1 2(h) = J61(h) = 0.

The following notation is convenient. If P
is a subset of Q and if h e HP(Q) then h I 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 X is a

compact Hausdorff space, if A is closed, and if h e HP(X)

such that h I A = 0, then there is an open set U containing

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;

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

f* : HP(Y, B) HP(X, A).

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

ht : (X, A) -> (X x T, A x T)
is defined by ht(x, t) = (x, t),

then -K*r = ?h* for r and s any elements of T.

Definition 1.7. A space X is unicoherent if

and only if X is connected and X = A V B with A and

B closed and connected implies A \ B is connected. X

is hereditarily unicoherent if every subcontinuum2 of X

is unicoherent.

Theorem 1.15. If X is a continuum and if

H1(X) = 0 for G t 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 F C A is a floor for h if and only if h I F 6 0

while h I F' = 0 for any closed proper subset F' of F.

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

Theorem 1.17. If (X, A) is a compact pair, if
R = A x A U A where A = { (x, x) I x E X } then
HP(X, A) HP(X/R) for all p> 1.




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 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 turn s 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.

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


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-

codimensional compact connected and locally connected

topological semigroup with unit and zero is a tree.

L. W. Anderson and L. E. Ward, Jr. in 1961 [1] 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].

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


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) H1(Rx) = 0 for every x in X,

(iii) the collection { RxJ 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

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 [171

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)


RX= J {Rx I 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 H1(RX) = 0. If there were a

non-zero h E Hl(RX), then there would be a maximal (non-

void) tower 3 of closed subsets A of X such that

h IRA # 0. Let A= { A I A Then

hi RAo :j 0, for if h I RAo = 0, then by the Reduction
Theorem (Chapter 1, Theorem 1.12.) there woulcdi'be an open
V ) RAo such that hi V* = 0. It would then follow
from Theorem 1.1. that R[-1] V would be an open set
containing A If R1-1]V is designated by U then
R U ( V so that there is an A in 7 with A ( U and
RA C RU ( V*; therefore h I RA = 0, a contradiction.

Case 1. Card Ao = 1, i.e., Ao = x By
(ii) H1(RA ) = 0, a contradiction.

Case 2. Card Ao > 1. Write A = A1J A2
where both A1 and A2 are proper closed subsets of A .
We consider the following part of the Mayer-Vietoris exact
sequence (Chapter I, Theorem 1.10.),
HO(RA 1 n RA2) 1(RA) H1(RA1) X H1(RA2).

Since by (iii) and (iv)
RA1 r\ RA2 = J { RA ^ Rb I (a, b) e A, x A21 is
connected, then A = 0 (Theorem 1.11.), and
h RA E Ker J = ImA = 0,
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 Hl(K) = 0 for every closed
subset K of RX [21], and thus every subcontinuum of RX
is unicoherent.

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 J Q would be a subcontinuum of RX which would not be

unicoherent, since P ri 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


(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 Hl(px) = 0 for every x in X,

(iv) {Px J 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

tree, and that P be a cutpoint-order.

Proof: We first prove the sufficiency.

Conditions (ii), (iv) and the first half of (iii) imply


Pa n Pb = J { Pxl x e Pa n Pb )

is connected, and thus Lemma 2.2. yields that X is a


Since X is compact and { Px I x E X} has the

f.i.p., then

{I Pxx X } 0

Indeed, it is a single point, the unique P-minimal element

of X. Let us denote by 0 the set

Px I x X }

We prove that P = Q(0). If (a, b) E 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(O) C P.

Conversely, if (a, b) is in P, then since a is in

aP r\ Pb, and since both aP and Pb are continue, then

aP vk- Pb is a subcontinuum of the tree X, and therefore

2. Condition (v) is not necessary for X to be
a tree.

3. Only reflexivity and transitivity of P are


by Lemma 2.1. it is unicoherent. Thus aP r, Pb is also a

continuum. Now, by virtue of the Hausdorff-i.4Lxima1lity--

Principle, aP n 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 E B. The set A contains a maximal element m.

Define A' and B' by the equations

A' = Pm r\ C and B' = C Pm.

B C mP, and since A C A', then B' C B. Now

A' \ B' C Pm n (mP n B) = (Pm n mP) \ B = ,


C = A' V B'

is a separation. If b0 designates the minimal element

in B', then by the maximality of C

mP r" Pb0 = { m, bo )

which contradicts the connectedness of mP n Pbo

Therefore, any maximal P-chain in aP r Pb is connected.

(c) aP rN Pb has a unique maximal P-chain,

which we denote by C p(a, b).

If C and C' were two distinct maximal P-chains

in aP r\ 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 E C C'

C n C' = (Px rA C r\ C') v (xP R C C') =

C A (Px u xP) n C'

is obviously a separation, a contradiction.
Since (0, b) E Q(O) 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 Cp (0, b)

the unique connected P-chain in Pb containir; 0 and b.

Since a Q-chain is also a P-chain, then

Cp(O, b) = CQ(O, b).

Similarly, there is a unique connected P-chain Cp(O, a)

in Pa containing both 0 and a. It is clear that

Cp(O, a) Q Cp(a, b) = Cp(O, b) = CQ(0, b).

As a consequence, a E CQ(0, b) and hence (a, b) E 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


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

HI(x) = 0. We now show H1(A) = 0 for every A = A* C X

and thus X is of one codimension, unless X '.is degenerate.

Suppose on the contrary that H1(A) 4 0 for some closed
subset A of X. If h is a non-zero member of HI(A),

then by the Floor Theorem (Theorem 1.16.) there is a floor

F Cj A for h, which is connected. The set F beirn a

subcontinuum of a tree is itself a tree and hence is acyclic.

Therefore, H1(F) = 0 which contradicts the fact that F

is a floor, and thus H1(A) = 0. The semilocal connectedness

of X follows from the fact that X is compact and locally


Proof of (ii). This is proved in Ward [24].

Proof of (iii). The cutpoint 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),' H1(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. (1) There is a component

A 0of A such that M does not separate A and B in

X. For if otherwise, to each component A? of A there is

a pair of disjoint open sets Gh and Th such that

X M = G\ U T and G? ) A?, T- B. Since A is
either compact or consisting of finitely many,components,

there is a finite subfamily {G1, G2, ..., GmJ of G'S

such that A C, { Gi. i = 1, 2, ..., m } Whence
m m
G = Gi and T = (\ Ti are two disjoint open sets

such that X M = G T and G ) A, T ) B so that the

hypothesis (i) is contradicted.
Throughout the rest of the proof, let f[S] be

the number of components of the space S. Let I = VI ? A EA

be the collection of all open sets Vh containing M but

missing A such that #[X VQ] is finite. For each

V E let R1 be the component of X V, that contains

the eCWijpoiKclLt Ao of A. Delignate R U ( RX N e A }

(2) R is open. Let y E R, and let V E -

be such that Rh contains y. By the normality of X

there is a V e ( with VC V V ( Since X V

has only a finite number of components and since

(X V )o ) X V ) V,

we have Rh ( R C. (R. Therefore R is open.

(3) For each V E there is an R, containing
R such that

# [R4 n (X Va)] = [ [R- (X Va)]

for all R containing R Let C1, C2, ..., Cn be the

n components of X Va, so that

X Va = C1 J C2 C ... n Cn

where C, = R., then for any Vg contained in Va we have
X Vo containing X Vo, so that each Ci must be either
totally contained in R., the component of X VP contain-
ing Ao, or disjoint from Rp. Thus

#[RI t (X V,)] # (X V,) = n.




# [R,6 n (X v)] < #[R? n (X Va)]
R. > R6 ) R Hence there is an R, ) Ra such that

number #[R, n (X Va)] is the maximal so that

#[R (x Va)] = #[R n (x Va)]
all R containing R .

(4) For each Va E Z there is an Rh

R L (X V,) = R- n (X V,)
for all R containing R,. By virtue of (3) al

is an Rh containing R suchthat

#[RI (X V,)] = #i[R; (X V(
for all R containing R,. We note, further, 1

R\ (X V) = RN n (X V,)
for all R containing R For if as in (3)
we express

such that

above, there

,) ]


X Va = C1 U C2 j ... Cn

where, without loss of generality, Ci is contained in Rh

for 1 1, 2, ..., k (k < n) and Ci n R = 0 for
i = k + 1, ..., n, then it follows from R containingg
Rh and
#[R \ (X V,)] = [R; n (X V,)]
that Ci r R4 = for i = k + 1, ..., n. And therefore,
R / (X Va)= C1 U C2 U ... Q Ck

= Rh ) (X V,)

(5) 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
t- missing y. By the normality of X there is a Vp

in t, such that Vg ( V* C V By (4) there is an Rh

containing Rp such that
R ^ (x vf) = R? n (x v )
for all R containing Rh. Furthermore, if U is an
open set about y, then U f R O* .0 For, if we designate
the set U r\ (X VP*) by W then y is in both R M
and W so that W R O0 and there is an R 6 such
that W n R6 0 Without loss of generality we may
assume that R, contains Rh. Then
SP W n R 6 W r\ R6 C (X VP)
w n R, r\ (x v)
Sn R (x-
Thus, U n Rh ) W n R-* L 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

RT r\ B 4 0 for some T ; because otherwise.'

X M= R U (X -M -R)

would be a separation.of X M between Ao and B, so

that (1) would be contradicted. We now conclude the

result by taking N = R T .



In the Symposium of General Topology and its

Relations to Modern Analysis and Algebra (Prague 1961),

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 < 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 (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-

(i) If z separates Pa and Pb in X then


Pz = z.

(ii) The set of P-minimal elements *is connected.

are equivalent (Theorem 3.4).

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. 16]. 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

Pxr\ K Q

Let y be any element in PxrK. Obviously x 4 y, so

that there is an open set U containing x whose closure

excludes y. If V = U ( K, then PV = V, and x is in

V*. Since

PVC (Pvf = v",

then PV*< U and hence y E Px C PV* C U*. This contra-

dicts the fact that y k UY. Therefore the set K is


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 V B. Since P is closed, then

AP and BP are closed and X = AP v BP. Furthermore, the

connectedness of X yields AP r\ BP 4 Thus AP n BP

has a minimal element, say t. Since

Pt = (Pt r\ AP) V (Pt r\ BP),

(Pt r AP) ^ (Pt BP) = Pt n (AP r BP) = t.

If we designate

C = (Pt (\ 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 n\ 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 CM= Pt \ AP and D = Pt n 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,

for each x in C we see that Px C C, and hence PC C C,

and it follows that

C* (PC)* 4 PC = P(C U t) : Pt ) D

which is a contradiction. Therefore K is connected.

Proof of Theorem 3.1. Let X z = A V B where

A and B are non-void separated sets and aP C A and

bP ( B. Let K be the set of P-minimal elements. Now,

if Pzp z, then K C X z and

K = (K r\ A) V (K r 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 V B)P = AP U BP.

And the hypotheses that X is connected and P is closed

yield AP r BP t Q Let z be a P-minimal element of

AP BP. Then there are elements a e A and bEB such

that {a, b} ( Pz z. The qualities
Pz = (Pz r- AP) V (Pz r-BP), and
Pz, (AP ^ BP) = z
show that
Pz z = (Pz r AP z) v (Pz r 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 3.5.

Example. Let X = { (a, b) \ a2 + b2 = 1 U
{(0, b) 1 1 < b < 2} for reals a and b. Let X be
endowed with the Euclidean topology. Let
P = { ((0, bl), (0, b2)) 1 1 <_ b <_ 2} U A

where A = { (x,x) I x X} Then P = P* is a both left
and right monotone partial order on X such that PAXC (PA)*
for each A ( X and
K = { (a, b) I a2 + b2 = 1} = S1 (1-sphere)
is the set of P-minimal elements.
It is to be noted in this example that HI(Px) = 0
for each x X while H1(K) H1(S) + 0 for any non-
trivial coefficient group. However the following equality

HP(X) 2 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) s 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 < Pu where K = { x I Px = x and

(iii) the quotient space X/K x K u A modulo

K X K V A is a topological semilattice under the natural

partial order induced by P, where A = 1 (x, x) I x E X ,

then, HP(X) n! HP(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) HP(X, K) M 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

takes X K topologically onto Y z. Thus'by the I.;ap
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 ft : (Y,z) -> (Y,z) by ft(y) = t A y for
all t E Y, where A is the semilattice operation on Y.
If for each t E Y define t: (Y, z) -> (YX Y, z x Y) by

At(y) = (y, t), then ft = A ? \t. Since by the Homotopy
Lemma (Theorem 1.14) A*= ? "*, thus

Since f is the identity map, so is f. Let
u z
i : (z,z) ( (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
fz = i h, the following diagram

f = f
HP(y, z) HP(Y, z)


HP(z, z)

is analytic, for all non-negative integers p, that is

f= f*= h*o i*. Since HP(z, z) = 0 for all integers
u z
p > 0 and HP(Y, z) = fu(HP(Y, z))( h*(HP(z, z)) for all

integers p > 0, then HP(Y, z) = 0 for all integers
p > 0 as desired.

(c) HP(X) t H(K) for all integers p > 0. By
combining the above results with the exact sequence for
the triple (X, K, 0 ) (Theorem 1.7), we obtain the exact

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 T : X X/KXKUA designates the

natural map, such that
(i) PA*( (PA)* for all A C X,
(ii) P is left monotone and Hl(v(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 KUA and
P= i (T(x), 7(y)) | (x, y) E P
then since X is compact and wT continuous, P is closed
and left monotone; indeed we have Pw(x) = vr(Px). By
virtue of Theorem 1.17, we have HP(X, K) S HP(") for all

p > 1. Since X is of codimension one we have' HP(X, K) =
0 for p >2 [21]. We show H1(X, K) = 0 by proving
H (x) = 0 Using the same argument employed in the proof
of Lemma 2.2 one sees that Hl(PS) = 0 for all closed sets
S in X; in particular we have HI () = 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) c HP(K) for all p L 1.
The equality HO(x) = HO(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) c HP(K) for all p > 0.
Proof. Let X, w and P be defined as in
Theorem 3.6. Since P is closed and 7rPx is a connected
chain P is closed and P7r(x) is a connected chain. It
follows then that each PTr(x) is a generalized arc and
hence HP(P7r(x)) = 0 for all p > 1. As it has been noted

in the proof of Theorem 3.6, HI(PS) = 0 for All 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) # 0 for some closed set A in X, then n would
have to be greater than 1, and A + D Let h be a non-
zero member of Hn(PA), then h I 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 0% of all open subsets U of A such
that h PU*= 0 forms an open cover of A. Also 0 is
closed under finite union, for if U1 and U2 are in ,
denote h h IPU1 U PU". Since P(U1 U UV P(U1 v U2)
-* 1 2 1 2=
= 1PU V PU to show G0 is closed under finite union it

suffices to show h = 0. In the following part of Mayer-
Vietoris exact sequence :
Hn-I(puI U,) A n JU n K,
H PU1 PU2) Hn(1PU VPU) 2 H (PU) X Hn(PU2)

since J(ho) = (h U, h i PU) = (0, 0), h is in the
image of A. But Hn (UIp PUP) = Hn- (PS) where S =
PUI PU is a closed subset of X, and hence IHn-(PS) =
0 by the minimality of n. This proves h = 0 so that
a is closed under finite union. Since A is compact,
A is a union of some finitely many elements of Thus
A must be in and so h = 0, a contradiction. This

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, C) [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 replacing the hypothesis (ii) by

(ii') each v(Px) is acyclic.

We conclude this chapter by exhibiting an e:.;;-iple

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) L (x, y) E S X S, xS V Sx C ySw 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 ar'c 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


Example. Let S = X U I be the subset of

Euclidean 3-space such that

X = { (x,0,0) j -1 < x < 1] V {(0,y,0) I -1 y < 1}

I= { (o,o,t) 0 o< t l ;

let S be endowed with the (0,0,I)
Euclidean topology and let (0,-l,0) (l,0,i)
the multiplication o on
S be defined below (the
usual multiplication of
reals is denoted by (-i,0,0) (0,1,0)
juxtaposition) :
i. Nop. = N for all \ E X and for all p E S,
ii. (I, o) is the usual semigroup of the real
unit interval; i.e.,

(0,O,tl)o(O;O,t2) = (O,O,tlt2),
iii. For each (O,O,t) E I and for any (0,y,0)EX,

(0,0,t)o(x,0,0) = (tx,0,O) and (0,O,t)o(O,y,O) = (O,ty,O).
Then, (1) (S, o) is associative.
(2) o : S S -> S is continuous.

(3) (S, o) is a clan with (0,0,1) 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-

t x+ y J7_
(x,y,t) It I
O 1 3


where -1 < x, y < 1, 0 < t < 1 and (tx)2 + (:xy)2 + (yt)2

= 0. The correspondence is indeed an isomorphism. From

this the assertions (1), (2) and (3) in the example

are self-evident.



A space X is said to have the f. p. p. (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 continue have the

f. p. p.

A space X is said to have the F. p. p. (fixed

point property for multifunctions) if every continuous

"multifunction1 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 continue X and Y both have the F. p. p. but their

Cartesian product 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 (Xh, hT, t\ ) 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).


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 continue as well.

Definition 4.1. The collection (X,, w A )
is as inverse system of spaces if:

(i) A is a directed set,
(ii) ? in A implies that X, is a Hausdorff

(iii) whenever A > j there is a continuous

function v : X> -> X, ,
(iv) if A > and 4 > v, then n- = IV7T

The function vn is called a bonding map. If A

is in A let Sh be the subset of the Cartesian product
P { Xx A e A } defined by

S = { x if ?A > j then wr\x(?) = x(u)} ,

where x(?) denotes the 7\-th coordinate of x.

Definition 4.2. The inverse-limit space X, of

the inverse system of spaces (X,, w, A ) is defined to
X= r{ S 1h E A

endowed with the relative topology inherited from the
product topology for P( X | A E A} ; in notation X
= lim (XA, 7 T,' A).

We write p, : P { X E A) X- for the A-th
projection of P { X E 1 A A} i.e., p,(x) = x(A) for all
x in P{ X | AI EA); the restriction phI X, will be de-
noted by 7rA which will be called a projection map. It is
readily seen from the definition that an element x of
P{ X I| A E A} is in X, if and only if "hrM,(x) = 7T(x)
whenever ?\ > t. A more detailed account of inverse limit
space may be found in Lefschetz [10], Eilenberg and Steenrod

[71, Capel [4] and Mardesic [11].

The following known results (see, e.g., [4], [101)
will be used.
Lemma 4.3. (i) The collection {(rIJ(UN)I ) E A

and UT is an open subset of XN J forms a basis for the
topology of Xe.
(ii) The inverse limit space X. is Hausdorff;
if N EA SN is a closed subset of P i X3 N E A} so
that X0 is closed in Pi X I A E A}
(iii) If XA is compact for each h in A then
X, is compact; if, in addition, each Xh is non-void
then X, is non-void.
(iv) If Xh is a continuum for each NA EA then
the inverse limit space is a continuum.

Lemma 4.4. If A is a compact subs et of X,

and if T =w k I w(A), then (I(A), 7( A) is an

inverse system of spaces such that A = lim (7ir(A), 7 A)

and each bonding map T is onto.

In the sequel, since we are only interested in

compact spaces, each projection map T- x will be assumed

to be onto; for if otherwise, by virtue of Lemma 4.4,

each Xh may be replaced by 7rx(X.) without disturbing

the resulting inverse limit space. We are now ready to

state our main result.

Theorem 4.5. Let (XV, T,' /A) be an inverse

system of compact spaces such that each Xh has the F. 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

(XX, 7wgT, A ) of compact spaces.

Lemma 4.6. If F : X -> X_ is a continuous
multifunction, define Fh : Xh X> by Fh = 7hF xT7 for

each X. Then Fh is a continuous multifunction.

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, P-l1(t) is a closed subset of XV. Thus,

Fh(t) is closed for every t in X
(ii) If CA is a closed subset of X., then
F lC.) is closed. It is readily seen that F-1 Vl(cC)

is closed in X~, and hence compact; therefore
'AF-"'A-(CA) = FA (CC) is compact and hence closed.

(iii) If UA is open in X,, then. F~I(U) is open.

7r and F being continuous, F-1 V N(U1 ) is an open subset
of X, It follows then, by virtue of Lemma 4.3 (i), that
VnF-IvF 1 (lu) = F1(U) is open.

Thus, by (i), (ii) and (iii) above, FA: X jXN
is continuous.

Lemma 4.7. Let F : X~ X, be a continuous
multifunction, let F, : Xh XN be defined as in Lemma 4.6.

Then, for each x in Xo,,
(i) (Fzw,(x), wA, A )2 and (7TF(x), 7T\, A )

are inverse systems of compact spaces,
(ii) lim (F 7,h(x), 7', A) = lim (7TwF(x) v,' /A),

(iii) F(x) = lim (FV?(x) 7hw, A).

Proof. Since each F, is continuous (Lemma 4.6)
and each Xh is compact, so is Fhr,(x) for all ?\ E A .

2. For simplicity in symbolism, henceforth if
AClim (X,, A r, A) then (rrA, 7r4, A) will mean
(vA, V A ~?h' A )'

To show that (Frn,(x), h*,, A ) forms an inverse system,
it suffices to show 7h4 Fh,(x) ( FT w(x) whenever \ > p.
To this end we first observe
7r^() C (V 7r )V,(X) = V-1

since vhLp = "r. From this we have

V CF() 7%^p.FQ7T ,i"(x)

V p.(7vrF 7w 1)V>-r1(x)

F= ( )F( p:w?) -1(x)

= vw pl7r j(x)

= F r (x),
by the definition of F,, Fp. and the equality n = T .P,

The fact that (w7F(x), p., 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 iT-hl1 (x) and hence,

7rF(x) C 7NF rr (x) = (wF 7rl)7-(x) = Fw(x).

rim (7hF(x), 7TT?, A) im (F7-j(x), wP, A).

To prove the other inclusion, we show
X..- lir (wrF(x), rp, A) C X, lim (Fh7T(x), Tp), A).
-Let y be in X lm (rF(x), ) then, by Lemma 4.4
Let y be in X, lir (7r F(x), 7 then, by Lemma 4.4
L y e X, -

there exists a L e A such that P(y) 4 V PF(x). Let UP

and VL be two disjoint open sets in X such that

(y) E U and 7rF(x) C V
so that
F(x)( C7r.1V P

It follows then from Lemma 4.3 (i) and the continuity of
F that there exists a 6 E A and an open set U6 in X8
such that x e 76 U6 and

(*) F(61U6) C v 7VP

Since A is directed, there is a E A such that
7 > p. and 0 > 6, we shall use this 0W throughout the
proof of this lemma. If we write Uo = 6U6 and use

the equality r = then (*) may be rewritten

as Fr U )

and hence

F oUo = r- F h(U ( 1.o)'V =-- ro(o7 -~

In particular,
FVoo(x) C ^o 1

Similarly, one obtains
.ro (y) E tf U

since vP,(y) E UP and wr, = AOi,.lo The fact that UP

and V are disjoint implies -l oV4 (% V 1 U = 0 and
P. 0.V ?\t 71)0p.U

consequently o, (y) J F ow o(x). From this we conclude

y lim (FaX,,(x), vh;\, A) as desired.

(iii) This follows immediately from (ii) and
Lemma 4.4.

Lemma 4.8. Let F : Xw X, be a continuous multi-
function, let Fh : X -> X- be defined as in Lemma 4.6 Let

E = {e I e X E X and eA E F,(eh)} then (E', wr,' A )

forms an inverse system.
Proof. It suffices to prove 7rhL E,( EL whenever

X > L. Let eX F,(e9), then

V (e.) E 7xF(e.) = 2Fr(e,) l(eF)

-i v-i : F l~e )
C 7rF(v-17T )7 A (e?) F.m7v_'(e\)

F= ( \A)1 (e,) =

Thus, "v E4 ( EL as is to be proven.

Proof of Theorem 4.5. Since each Xh has the
F. p. p. and by Lemma 4.6 each F, : X -> Xh is continu-

ous, each E is closed and non-void. By Lemma 4.8,
(Eh, iTv-, A ) is an inverse system of compact spaces, so
it has a non-void inverse limit space lim (E,\, FT7, A).
We now conclude the proof by showing that each x in
lim (E., Ay,' A) is a fixed point under F; i.e., xeF(x).

If x is in lim (Eh, 7r" A) then i,(x) e'Eh for all

A E A ; i.e., w,(x) E F.7r,(x) for all ? e A Conse-
quently, by Lemmas 4.4 and 4.7, we have
x = lim (wr?(x) ., A) E lim (Fh,(x) 7-r,,, A)

= 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 E F(x) we can make the following sharper

Theorem 4.5'. E = lim (E,, wI-, A).

Proof. From the proof of Theorem 4.5, we have
E ) lim (E,' 7r,, A).

It remains to prove that
E Cl1im (E?, IT^ 7,) A

Let x be in E, then x E F(x) and therefore, for all
E A ,

TTh(x) E v F(x) ( vF(Tl, ) (x) = F?(7r(x)).

That is, 7,r(x) E Eh for all X; consequently, by Lemma 4.4
E ( lim (E?, 7r,,, A).

A chain (UI, U2, ..., Un) is a finite sequence

of sets Ui such that Ui r U, f D if and only if
Ii J I 1. A Hausdorff space X is said to be chainable

if to each open cover LJ of X there is a finite open
cover U = (UI, U2, ..., U n) such that (i) U refines

1; (ii) ;J= (U1, U2, ..., 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

[131 that a bounded closed interval of the real numbers has
the F. p. p. imply the following result of Ward [251 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 [251.


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Shwu-yeng Tzeng Lin was born May 11, 1934 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

1960. In the fall of 1960 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.

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

Dean, Graduate School
Su pervisory Comittee:

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