SET FUNCTIONS AND LOCAL CONNECTMTY

 

Thesis fer the Deflree of Ph. D.
MECHEGAN STATE UNWERSWY
EUGENE LEROY VANDENBOSS

1970

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SET FUNCTIONS AND LOCAL CONNECTIVITY
presented by

Eugene Leroy Vandenboss

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Ph.D. degree in Mathematics

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ABSTRACT
SET FUNCTIONS AND IDCAI CONNECTIVITY

BY

Eugene Leroy VandenBoss

This is a study of the closure, Y, with resPect to con-
tinua with connected interior. Chapter one develops the ele-
mentary properties of Y; Chapter two deve10ps the basic re-
lationships between Y and T, where T denotes the closure
with reSpect to continua; Chapter three develops relationships
between Y and monotone maps.

In Chapter one the usual hypothesis is that S is a
compact Hausdorff Space. The main theorems are:

S Elocalll connected Eflflgfo}; ACS
Y(A) = C1(A).

S ‘13 locally connected 1£.§Eé.flflll.i£

1- Y(p)={p} £9121; 968 2:19

2. S i§_Y-additive.

 

 

;£_ S ii Y-symmetric, then S ‘13 locally connected at
p, i_ and only l__ Y(p) = {p}.

‘1: C $3 5 subcontinuum g: the continuum S, then
Y(C) _i_8_ a continuum.

In Chapter two S denotes a Hausdorff continuum. The

main theorems are:

f S is weakly irreducible, then S is locally con-

 

nected at p, if and only if. S is connected Im Kleinen at p;

Eugene Leroy VandenBoss

moreover, i_i: S _i_s_ also yosyndetic, then S is locally con-
nected.

S is weakly irreducible _i_f_ and _qn_ly _i_f_ for any sub-
continuum, W, _E S, S - W by: a finite number 9_f components.

S i_s loc&l_ly connected giggling E S X S .13
Y-additive.

_g ACInt(B) ch8 Eli T(B) =B, £1131 Y(A) cB.
(Here S need not be connected.)

This last result generalizes the theorem which states:
S is locally connected if and only if S is connected Im
Kleinen.

In Chapter three S denotes a compact Hausdorff space,
which need not be connected. The main theorems are:

_Le_t_ f begmonotone mapgg S Q52 Z,_t_l_1e_n
Y(f‘lom c Flora») :2; 92 A c 2-

_IEE f 19.29. open monotone majgi S 9339 Z, M
Y(A) == f(Y(f-1(A)) g2; §1_1 Ac: 2.

Let f _b__e_a_n open monotone mapo 8 gn_tg Z. $113.9

1. _Ij S _igY-additive, then Z ii Y-additive

2. f S _i_sY-synmetric, then 2 g Y-symmetric.

SET FUNCTIONS AND IOCAI CONNECTIVITY

By

Eugene Leroy VandenBoss

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1970

Q ~ (55%:

/ - 3112' ’7/

ACKNOWLEDGMENTS

I wish to thank Dr. H.S. Davis for his guidance and
time in the preparation of this work, also for his aid in
helping me form a style of writing. Above this I wish to
thank Dr. Davis for his time spent developing my mathematical

and teaching abilities.

ii

Chapter
1
2

3

TABLE OF CONTENTS

Page
E IEWNTARY m0 PERT IE S OF Y I O O O 0 O O C O O O 0 O I O O O O I 0 O O O O 1
REIATIONSHI m BENEEN Y MD T O O O O O O O O O O O O O O O O O O O 11

Y MDMONOTONEMB 0.000.000.0000...00.00.000.000. 20

BIBLIOGRAHIY 0...... ..... 0.... OOOOOOOOOOOOOOOOOO O... 26

iii

CHAPTER I

EIEMENTARY PROPERTIES OF Y

Definition. Let S be a set and let P(S) denote the
collection of all Subsets of S. Let S be a topological Space
and let ”‘3 P(S). e is the closure with resPect to n if and
only if e: P(S) « P(S) by the following rule: x is not an
element of 9(A) if and only if there exist N and element
of n Such that x is an element of the interior of N and
N and A are disjoint.

The following are directly verifiable.

Formulas. Let S be a topological space, n<: P(S)
and let 9 be the closure with respect to n, Ehgg

i. AC 9(A)

ii. 9(A n B) C 6(A) n 9(8)

111. 9(A) u 9(3) c: 9(A U B)
iv. 9(3) 8 S
v. 9(A) is closed
vi. If Ac: B, then 9(A) c: 9(B).

In [1] the set function T was defined and its basic
properties were discussed. T is the closure with respect to
the collection of continua.

This paper is a study of the closure with reapect to

continua with connected interiors.

Definition. Let S be a topological Space and let W

 

be a subset of S, then W is called a strong continuum if and

 

and only if
1. W is a closed compact connected subset of S
2. The interior of W is connected.
The set-function under consideration is defined as follows:

Definition. Let S be a topological Space, then Y

 

is the closure with reSpect to the collection of strong continua.
Definition. Let S be a topological space and let p
be an element of S, the 77((p) is the set of all strong continua
W in S such that p is an element of the interior of W.
Following are some immediate results for all topological
spaces S.
Theorem 1. I££_ W CZS be a strgngicontinuum.then the
closure of the interior of W is a strong continuum of S.

Theorem 2. Let A be a subset of S, then

 

Y(A) = {xwemm = w n A 34 ,5} u {x|W(x) = ,5} = {x|we'm(x) =w n A # go}.

A sequence of related set-functions can be defined as
follows.

Definition. Let n be a positive integer and let A
be a subset of a topological Space S, then Y1(A) = Y(A) and
Ynflm = vane».

Formulas for Y.

1. ACY(A)
2. Y(A.n B) CLY(A) n Y(B)
3. Y(A) U Y(B) CY(A U B)

a. Y(S) = s

5. ‘Y(A) is closed
6. If ACB then Y(A) CY(B)
7. Let m s n then Ym(A)<:'Yn(A).

The following example shows that 77((p) can be empty
and that the inequality in formula 1 may be proper for closed
sets.

Example 1. Let S = {(%,O)}n is a positive integer} U
{(0,0)} with the relative plane topology, then. WK(0,0)) = ®
and Y((fi30)) = {(%,0), (0,0)}.

The following example shows that the inequality in
formula 1 may be proper for a closed set when 1m(p) # ¢ for
all p E S.

Example 2. Let S be the Subcontinuum of the plane
defined by the union of the closed line segments between (0,1)
and (fiyO) for n 2 l and the closed line segment between
(0,0) and (0,1); t0pologically this is the cone over example 1.
In S, Y((O,l)) is the closed line segment between (0,0) and
(0,1).

The following example shows that formula 3 need not be
an equality.

Example 3. Let S be the Subcontinuum of the plane
defined by the union of closed line segments between (0,1)
and q%,0) for n 2 l, the closed line segments between {0,-1)
and %,O) for n 2 l and the closed line segment between
(0,1) and (O,-l); this is t0pologically the suspension over
example 1 with vertices (0,1) and (0,-l). In S, Y((O,l)) =

(0,1) and Y((O,-l)) = (O,-l) but Y({(O,l), (O,-l)}) is the

closed line segment between (0,1) and (O,-1).

The following example shows that Y(A) need not be
Y2(A) and similar examples can be found which have the property
that for m and n distinct positive integers Ym(A) need
not be Yn(A).

Example 4. Let S = {(x,% + % sin(fi))\0 < x s l} U
{(x,-% +>k sin(%))}0 < x s 1} U {(O,y)}-% S y s k} with the
topology induced by the plane, then Y((O,%)) = {(o,y)Io s y s s}
and quoe» = {(0.y)I-% s y s a}.

Definition. A Space S is called Y-additive if and

 

only if for any collection {Ad} of closed subsets whose union
is closed Y(UanD = U{Y(Aa)}.

The Space in example 3 is not Y-additive since
Y({(O,1)}) U Y({(O,-1)}) #‘Y({(O,l), (O,-1)}).

For the remainder of this chapter S will denote a
compact Hausdorff Space.

Theorem 3. Y(¢) = ¢ if and only if S has a finite

 

 

number of components.

 

Proof. Let S have a finite number of components. Then
each component is both open and closed, hence IW(P) ¢ ¢ for all
p E S and thus if p E S then p E Y(¢).

Let Y(¢) = ¢. Then 77((p) 35 (15 for all p 6 S, hence
each component of S is open. Since S is compact and each
component of S is Open, S has only a finite number of com-
ponents.

The following theorems Show the relationships between

the concept of locally connected and the set function Y.

Theorem 4. S is locally connected at appoint p if and

only if for all subsets A p£_ S, if p is an element of Y(A),

 

£232. p is an element of the closure of A.

Proof. Let S be locally connected at p and suppose
p is not an element of the closure of A. There exists an
open set U such that p 6 U and Cl(U) n A = ¢. Since S
is locally connected at p, there exist an open connected set
V such that p E V C U. Cl(V) n A = ¢ and Cl(V) 6771(p).
Therefore p is not an element of Y(A) and it follows that
if p is an element of Y(A), then p is an element of the
closure of A.

Let p be an element of S such that for all Ac: S,
if p is an element of Y(A), then p is an element of the
closure of A. Let U be an open set containing p, then
S - U is a closed set and p is not an element of S - U.
There exist W 6 771(p) such that W n (S - U) = ¢, hence
p E Int(W) CiW CiU and thus, S is locally connected at p.
The theorem is proven.

Corollary 5. S is locally connected if and only if
£2£_ A<: S, Y(A) = Cl(A).

The next theorem shows the relation between Y-additivity
and the locally connected Spaces.

Theorem 6. S is locally connected if and only if

1. Y(p) = {p} for all p E S 33d
2. S i§_Y-additive.
Proof. let Y(p) = {p} for all p E S and let S

be Y-additive. Let A¢:.S, then Cl(A) CZY(A) CZY(C1(A)) =

Y(U{{P}IP e cum)

Cl(A). Hence Y(A)

U{Y(P)IP E Cl(A)} = UIIPIIP E Cl(A)} =

Cl(A) and thus by corollary 5 S is
locally connected.
Let S be locally connected and let {Au} be a set
of closed sets such that UIAQ} is closed. By corollary 5
Y(Aa) = Ad and Y(U{Aa}) = U[Aa}. Hence U{Y(Aa)} = U{Aa} ._-
YqJ{Aa}) and S is Y-additive. Since 8 is Hausdorff, for
p E S {p} is closed and by corollary 5 Y(p) = {p}. The
theorem is proven.
The following two examples Show that neither Y-additivity
or Y(p) = {p} for all p E S implies the other.
In example 3 S was not Y-additive, but Y(p) = {p}
for all p element of 8 (hence S is not locally connected).
In example 2 S was Y-additive, but Y((O,l)) =
{(O,y)‘0 s y s 1} (hence S is not locally connected).
Definition. S is called Y-symmetric if and only if
for any two closed subsets A and B of S, if Y(A) is
disjoint from B, then Y(B) is disjoint from A.
Theorem 7. LEE S pg_Y-Symmetric, then S i§_Y-additive.
Proof. Let {Au} be a set of closed sets such that
U{Aa} is closed, then U{Y(Aa)} CiY(U{Ad}). Hence all that
needs to be shown is that YOJ{Ad}) C:U{Y(Ad)}. Let S be
Y-Synrnetric. Let p e Y(U{Aa}), then Y(U{Aa}) n {p} 39 ¢,
hence Y(p)r1 GJ{Aa}) # ¢. Hence there exists 5 such that

Y(p) n A # ¢. Therefore {p} 0 Y(AB) f ¢, hence p E Y(AB)

B
and therefore p E U{Y(Ad)}. Therefore Y(U{Ad}) c:U{Y(Aa)}

and the theorem is proven.

In example 2 S is Y-additive but S is not Y-symmetric
Since Y((O,l)) = {(o,y)\o s y s 1} 3 {(0,0)} and Y((0,0)) =
{(0,0)}.

Theorem 8. L3; S pg.Y-symmetric, then S is locally

connected at p, if and only if Y(p) = {p}.

 

Proof. Let S be Y-symmetric and locally connected
at p. Let q 6 Y(p), then Y(p) n {q} i ¢, hence Y(q) n {p} # ¢,
hence p G Y(q); since 8 is locally connected at p,
p e Cl({q}) = {q} and hence p = q. Therefore {p} = Y(p).
Let S be Y-symmetric and {p} = Y(p). Let U be
an Open set containing p. If p E Y(S- U), then (S - U) n
Y(p) # ¢, but Y(p) = {p} and hence (S - U) n {p} # ¢, a
contradiction. Therefore p d Y(S - U) and hence there exists
W 67((p) such that W n (S - U) = ¢. Thus p 6 Int(W) CW CU
and the theorem is proven.
Following the convention in [4] page 6, 8 is called
a filter-base in a topological space S if and only if
1. ch(S)
2. 3 ¢ ¢
3. A,B E 8 implies that there exist C E 8 Such
that C is a subset of A intersect B.
8 is said to be proper if and only if ¢ 4 3.
g is Said to be closed if and only if A E 3 implies
A is closed.

Theorem 9. If 8 is a prgper closed filter-base in S,

the; YmIAIA 6 3}) = nIY<A>IA e :5}.

-An

 

'Fr'

Proof. Let x 6 Y(fl{AIA 6 3}), then for all W E 77((x)

W n (n{A}A E {5}) 9‘ ¢. Therefore for all A E :5 and all W E 772(x),
W n A 9‘ ¢, hence x E Y(A) for all A E 8, hence
x e n{Y(A)IA e :3} and Y(n{A\A e 3}) cn{Y(A)|A e :5}.

Let x é Y(n{A}A E 8}), then there exists W E 771(x)
such that w n (n{AIA e {3}) = 95. Hence w c s - n{A\A g g}
and {S - A}A E 3} is an open covering for W. Since W is
compact there exist A1,...,An such that
w:u{s -AiI1 s i sn] =3 -n{AiI1 s i Sn}. Since {3 is a
prOper closed filter base, there exists an element, A, of 3
Such that Acanill s i S n}, hence W n A = 95 and x é Y(A).
Thereforex 4 n {Y(A)}A e g} and n{Y(A)\A e g} c: Y(n{A}A e 5}).
The theorem is proven.

Theorem 10. S i_sY-additive if and only_if for each
£33 A 93‘}. B of closed subsets of S Y(A U B) = Y(A) U Y(B).

Proof. Let S be Y-additive and let A and B be
closed subsets, then Y(A) U Y(B) = Y(A U B).

Let Y(A) U Y(B) = Y(A U B) for any two closed subsets
of 8. Let {AaIa E d} be a set of closed sets such that
U{AaIcy E d} is closed.

Since U{Y (Aa)Io E d} C Y(U{Aa}a 6 4}) all that needs
to be shown is that Y(U{Aa\or E.d}) C U{Y(Aa)}a E a}.

For each or 6 a let 3(Aa) be the collection of closed
subsets B of S such that Ad C Int(B). If A0 = q) then
Y(Aa) = n{Y(B)|s 6 {50.0)}. If Ad 3‘ ,3, then 39.0!) is a closed
prOper filter base of S and Since n{B‘B E 8(Aa)} = A0,

Y(Aa) = n{Y(s)\B e (Satan.

 

‘l‘i‘tl‘fll‘ll‘ll I'll I III

Suppose x i U{Y(Ad)Io E afl. Then for each a €<7
there exists Ba e {3(Aa) such that x 4 Y(Ba). {Int (Ba)Ior 6 a}
is an open covering of the compact set U{AdIo E a? and hence
there exists B1,...,Bn such that U{Aa\a e a} c u{1nt(si)|
l s i g n}. By hypothesis Y(U{BiIl s i s n}) =
U{Y(Bi)I1 s i s n}, hence x t Y(U{Bi}l s i s n}) :3
Y(U[AaIa e m). Therefore Y(U{Aan e a}) c U{Y(Aa)Io E d}
and the theorem is proven.
The following example shows that Y({p,q}) = Y(p) U Y(q)
for any two elements of Z and that Z is not Y-additive.
Example 5. Let S be the Space in example 3 and let
Z = S X I. Then Y(p) U Y(q) = Y({p,q}) for all p,q E Z,
but Y((0,1) x I) = (0,1) x I and Y((0,-1) x I) = (O,-1) x I
but Y(((O,l) x I) u ((o,-1) x 1)) = {(o,y)|-1 s y s 1} x 1.

Theorem 11. If for all p E S and any finite collection,

 

 

 

{W1}, of elements of 77((p), there exists W E 77((p) such thc'3_t_
W CZHIWi}, £322. 8 i§_Y-additive.

Proof. By Theorem 10 all that is needed to be shown is
that Y(A) U Y(B) = Y(A.U B) for any two closed subsets of S.
By formula 3 all that needs to be shown is that Y(A U B)<:
‘Y(A) U Y(B).

Let A and B be any two closed subsets of S and
let p d Y(A) U Y(B). There exists W1,W2 €1m(p) such that
W1 0 A = n = W2 .1 B. By hypothesis there exists W E 77((p)
such that W CW1 0 W2. Hence W n (A.U B) = ¢ and
p 4 Y(A U B), thus Y(A U B) C1Y(A) U Y(B). Compare with

Chapter 1, paragraph 2 of [5].

10

Theorem 12. Let C be a Subcontinuum of the continuum

S, then Y(C) is a continuum.

 

Proof. Since Y(C) is closed and hence compact, all that
needs to be proven is that Y(C) is connected.

Suppose Y(C) = A U Bsep and C CIA“ Since 8 is normal,
there exists U and V Open set with disjoint closures such
that ACU and BCV. Since CCACU and B CV, Y(C)
is disjoint from Fr(V), where Fr(V) denotes the boundary of
V. Therefore for all y 6 Fr(V), there exists Wy E 77((y) such
that Wy n C = ¢. Since {Int(Wy)} forms an open covering of
Fr(V), there exists strong continua 'W1,...,Wn such that
wi n c = C, and U{Int(Wi)|1 s i s n} :> Fr(V). If K is a
component of V then Cl(K) n Fr(V) # ¢, hence
V U(U{Int(Wi)}1 s i s n}) has only a finite number of components.
Therefore each component of V U (U{Int(Wi)\l s i s n}) is Open.
Let b E B and Kb be the component of V U (U{Int(Wi)}1 S i S n})
containing b, then Cl(Kb) E WKb) and Cl(Kb) n C = ¢. There-
fore b 4 Y(C), but this contradicts the fact that B C Y(C).

Therefore Y(C) is a continuum.

CHAPTER 2

RELATIONSHIPS BETWEEN Y AND T

This chapter develOpS some relationships between T
and Y.

Definition. Let S be a topological space, then T
is the closure with reSpect to continua.

Theorem 13. Let A be a subset of the topological

 

gpggg S, £§gp_ T(A) is a subset of Y(A).

Proof. Suppose x 4 Y(A). Then there exists W E 77((p)
such that W n A = ¢. Since W E m(p), x E Int(W) and there-
fore x d T(A). Hence T(A) C Y(A).

The following example shows that T(A) need not be
Y(A).

Example 6. Let S be the Subcontinuum of the plane
defined as follows. S is the closed line segments between
(1,0) and (O,%) for n 2 1 union with the closed line
segments between (l,%) and (2,0) for n 2 1 union with the
closed line segment between (0,0) and (2,0). T((2,0)) =
{(x,o)\1 s x s 2} and Y((2,0)) = {(x,0)\o s x s 2}.

The following is an example of a Space that is Y-additive,
but not T-additive.

Example 7. Let S be the closed line segments between

0
(O,%,O) and (1,0,0), between (l,%30) and (2,0,0), between

11

12

l
(2:;

(2,-%,0) and (1,0,0), between (3,-%,0) and (2,0,0), between

,0) and (3,0,0), between (1,-§,0) and (0,0,0), between

(1%,O,%) and (0,0,0), between (3,0,0) and (12,0,fi) for
n 2 l unioned with the closed line segment (0,0,0) and
(3,0,0). T((0,0,0)) = [(x,0,0)\0 s x s 1} and T((3,0,0)) =
{(x,0,0)\2 s x s 3}, but r({(0,0,0), (3,0,0)}) =

{(x,0,0)\0 s x s 3}.

Let L = {(x,0,0)\0 s x s 3}. If A.C:S is closed
then Y(A) = A if A n L = ¢ and Y(A) = A u L if A n L # ¢.
Thus S is Y-additive.

For the remainder of this chapter S will be a compact
Hausdorff continuum.

Definition. S is called weakly irreducible if and

 

 

only if given C1,C2,...,C subcontinua of S,

n
S - UICiIl s i s n} has a finite number of components.
Lemma 14. Let S be weakly irreducible and let C

be a Subcontinuum of S, then Int(C) has only a finite number

 

of components.

 

Proof. Let S - C have components K ,...,K , then
1 n
Int(C) = S - U{C1(Ki)}l s i S n} which has only a finite number

of components.

 

 

Theorem 15. Let S be weakly irreducible and let A

be a subset of S, then T(A) = Y(A).

 

Proof. Since T(A) C:Y(A), all that needs to be shown
is that Y(A) C1T(A). Let x é T(A), then there exists a con-
tinuum W such that x 6 Int (W) and W n A = ()5. Since Int (W)

has only a finite number of components each component is open;

13

let K be the component containing x, then Cl(K) E‘m(p)
and Cl(K) n A = ¢, therefore x é Y(A); hence Y(A)<: T(A)
and the theorem is proven.

Corollary 16. '_f S is weakly irreducible, then

1. S is Y-symmetric

2. S is locally connected at p, if and only

 

if_ S is connected Tm Kleinen at p.

La.)
0

H
I'h

S is also aposynedetic, then S 25
locally connected.

Proof. This follows from the previous theorem and from
Theorem 6 of [2].

The following develops a weaker statement which is
equivalent to weakly irreducible.

Definition. Let S be a continuum and let A be a
subset of S. S is called irreducible about A if and only
if for C a subcontinuum of S such that A ClC, then C = S.
Notation S = [A].

Definition. Let S be a topological space, let A and B
tuztwo disjoint closed subsets of S and let M be a Subcontinuum
of S. M. is called irreducible from A pp. B if and only if
M intersects both A and B non-voidly and no prOper sub-
continuum Of M intersects both A and B.

The following two theorems are from [6].

Theorem A (Theorem 43). Lg£_ A 22g, B be two disjoint

closed subsets 2;. S, then S contains 3 continuum irreducible

from A _t_9_ B.

14

Theorem B (Theorem 47). Let A and B be two disjoint

closed subsets of S and let M be an irreducible continuum

 

 

from A pp. B, then M - (A.U B) and M - A are connected.

 

Theorem 17. Let C1,C2,...,Cn be disjoint subcontinua

 

 

pf S, then there exists a component K pf S - U{Ci|l S i S n}
such that Cl(K) n 01 i‘ (a gig C1(I()n (U{ci|2 s i s n}) ié (2).

Proof. Since C and U{Ci}2 S i S n} are closed dis-

1
joint subsets of S, 8 contains a continuum, M, irreducible from

C to U{Ci}2 S i S n}.

1

Let L = M - U{ci\1 s i s n}, then Cl(L) n 01 # (b,
Cl(L){1 QJ{Ci}2 S i S n}) # ¢ and L is connected. Let K
be the component of S - U[CiI1 S i S n} containing L, then
Cl(K) n 01 59 ,5 and 0100 n (\J{Ci\2 s i s n}) 9‘ $-

Corollary 18. Let C1,C2,...,Cn be disjoint subcontinua

igf S, then there exists K1,...,Km, components of

 

 

S - U{Ci|1 S i S n}, m S n, such that (U{K1I1 S 1 S m}) U

 

{U{Ci\1 S i S n}} is a subcontinuum of 8.
Theorem 19. S is weakly irreducible if and only if

given any W a subcontinuum of S, S - W has a finite number

 

of components.

 

Proof. Let S be weakly irreducible and let C be a
subcontinuum of S, then S - C has a finite number of components
by definition.

Let S be Such that for any subcontinuum W of S,

S - W has a finite number of components. [at C1,C2,..-,Cn

be subcontinua of S. Then U{Ci|l S i S n} =lJ{Mi}l S i S m}

where the M1 are disjoint components of U{Ci|1 S i S n}.

15

The M1 are disjoint subcontinua of S. Therefore there exist
k1,...,kL components of S - U{Mi|l S i S m} such that
c = (U{ki\l s i s 5}) u (UMi‘1 s i s m) is a continuum. There-
fore S - C has a finite number of components t1,...,t0. There-
fore S - U{Ci}1 S i S n} has less than or equal to L + 0
components. The theorem is proven.

Theorem 20- L35 3 = [{x1,x2,...,xn}], then S is weakly
irreducible.

Proof. Let C be a subcontinuum of S and let ki

be the component of S - C containing x if x1 E C.

i
(U{kiI1 S i S n and xi 6 0}) U C is a subcontinuum and
{x1,x2,...,xn}<: 0J{ki}1 S i S n and x1 i C}) U C; therefore
3 = QJ{ki|l S i S n and xi 4 C}) U C. Hence S - C has less

than or equal to n components. Therefore 3 is weakly

irreducible.

Corollary 21. f S = [{x1,x2,...,xn}], then

1. S is connected Im Kleinen at p if and only

 

.if S is locally connected at p
2. S is Y—Symmetric.

Proof. This follows from Corollary l6 and Theorem 20.

In Example 2 S = [{ (%,O)}n > O} U {(0,0), (0,1)}],
but S is not weakly irreducible since S - {(0,0)} has an
infinite number of components.

Definition. B is called a compact separator of the
topological Space 3 if and only if

1. B is compact

2. S'B=HUKSEP.

16

Theorem 22. if A is a subset of S, then Y(A) inter-

 

 

 

sects any compact separator of S that separates A from any
point of Y(A).

Proof. Suppose the theorem is false and there exists a
compact set B such that S - B = H U K sep, AC: H, p 6 Y(A) n K
and S - BID Y(A). Since B is closed, both H and K are
open and since Cl(K) n H = ¢, if k is a component of K then
Cl(k)!) B # ¢. Since B CZS - Y(A), there exists a finite number
of strong continua, Wi, Such that Wi n A 3 ¢ and B<:
U{Int(Wi)}. Therefore K U (U{Int(Wi)}) has a finite number
of components and each component is open. Let k be the
component of KU (U[Int(Wi)}) containing p, then Cl(k) €77}(p)
and Cl(k)[1 A = ¢. Therefore p 6 Y(A), contradiction. The
theorem is proven.

Corollary 23. f A.C:S, then evegy component of Y(A)

intersects A.
Proof. See Corollary 1.1 of [7].

Corollary 24. .l£ A is a closed subset of S and

 

Y(A) is totally disconnected, then Y(A) = A.
Proof. See Corollary 1.2 of [7].

Corollary 25. Let A and B be closed, totally dis-

 

 

connected subsets. whgre ACS1 and B c:82, then for any

closed subset K C A x B C S X 82, Y(K) = K = T(K).

1
Proof. See Theorem 2 of [7].

Theorem 26. Let p E S X S, then Y(p) = {p}.

Proof. Let q = (a,b) and p = (c,d) be two distinct

points of S x S, then a # c or b i d. It may be assumed

17

that a # c. There exists U and V open set such that a E U,
c G U. and d é V: Then (a,b) is an element of the open set
(U X S) U (S X V) and (c,d) is not an element of the closed
set (Cl(U) X S) U (S X Cl(V)).

To Show that (U X S) U (S X V) is connected, it need
only be shown that for (e,f) and (h,g) elements of (U X S) U
(S x V) that there are elements of a connected subset of
(U X S) L) (S X V).

Case 1. (e,f) and (h,g) are elements of U X S or
S X V. It may be assumed that there are elements of U X S.
Let t E V then ({e} X S) U ({h} X S) U (S X {t}) is a
connected set Since ({e} X S) n (S X [t}) ' (e,t),
({h} XS)n (S x {t}) = (n,t) and ({e} XS)U ({h} XS) U
(SX[t})C UXSUSXV.

Case 2. (e,f) E U X S and (h,g) E S X V then
([e} X S) U (S X {g}) is a connected Subset of (U X S) U
(S X V). The theorem is proven.

Corollary 27. The followipg are equivalent

1. S is locally connected

2. S X S i§_Y-additive

 

3. S X S is T-additive.

 

In the following theorem the hypothesis that S is
connected is not necessary.

Theorem 28. I_f_ ACInt(B)CBCS a_n<_i_ T(B) =B,
£1193 Y(A)CB.

Proof. First we show that each component of S - B

is Open. Let K be a component of S - B and let x E K.

18

Since x i T(B), there exists a continuum. W such that
xélnt(W) and WnB =¢. Hence WCK and x EInt(W),
therefore x e Int(K). Since x could be any element of K,
K is Open.
Now we Show that Y(A) ClB. Let p E S - B and let
K be the component of S - B containing p. Then Cl(K) €1m(p)
and Cl(K)f] A = ¢, therefore p G Y(A) and thus Y(A)<: B.
Corollary 29. S is locally connected if and only if

S is connected Im Kleinen.

 

Proof. If S is locally connected then S is connected
Im Kleinen.
Let S be connected Im Kleinen, then T(A) = Cl(A)
and S is T-additive [2]. Let U be an Open set containing
A, then T(C1(U)) = Cl(U), hence Y(A)<: Cl(U) by Theorem 28.
Cl(A) C Y(A) Cn[C1(U)‘A C U and U is open} = Cl(A), hence
T = Y. Therefore S is Y-additive and Y(p) = {p} for all
p E S, thus S is locally connected.
Davis has the following theorem in [9].
Let S be a compact Hausdorff Space, then the following
are equivalent
1. T(A)}fi B = ¢ where A and B are closed
2. There exist closed subsets M and N Such that
AC Int(M), B C Int(N) and TM) 0 N = ¢.
Corollary 30. If_ T2(A) = T(A) for all A CZS, £233
T(A) =‘Y(A) for all Ac: 8.
Proof. Let x Z T(A), then there exists an open set U

such that A<: U and x i T(Cl(U)L [9]. Since T(T(Cl(U))) =

19

T(Cl(U)) and Since A C U C Int (T(Cl(U))), Y(A) C T(Cl(U)) by

Theorem 28, therefore x i Y(A). Thus T(A) = Y(A).

CHAPTER 3

Y AND MONOTONE MAPS

This chapter is a study of the relations between Y
and monotone functions.

For the following theorems all Spaces are Hausdorff.

Definition. A function f from S onto Z is called
monotone if and only if f is continuous and f-1(z) is
connected for all z E Z.

Theorem. L55 f be an open monotone map of S 2353
Z and let A be a connected subset of Z, Epgp f-1(A) is
connected.

Proof. See Chapter VI, section 3, problem 1 of [8].

Theorem 31. Let f be a closed monotone map of S
pppg Z and let A be a connected subset of Z, ppgp f- (A)
is connected.

Proof. Suppose the theorem is false and let C be a
connected subset of Z such that f‘1(C) = M.U N sep.
f(M) n f(N) = 0: for if p e f(M) n f(N), then r'1(p) n N a! 0
and f-1(p) n M # ¢, but f-1(p) is connected, hence
f-1(C) # M n N sep. Therefore f(N) n f(M) = q).

C = f(f-1(C)) = f(M U N) = f(M) U f(N) and C is
connected. Therefore Cl(f(N)) n f(M) ¢ ¢ or f(N) n Cl(f(N)) ¢ ¢.

We may assume that Cl(f(N)) n M # ¢. Let p E Cl(f(N)) n f(M),

20

21

then f-1(P)I] M # ¢. f-1(p) is connected, hence f-1(p)<: M.
{p}<: Cl(f(N)) = f(Cl(N)) Since f is a closed map. Therefore
f-1(p)[1 Cl(N) # ¢, hence Cl(N)t] M # ¢. This is a contradic-
tion of separability. Thus f-1(C) is connected.

For the following theorems all Spaces are compact
Hausdorff Spaces.

Theorem 32. L33 f be a monotone map of S 2353 Z
and let W be a Strong subcontinuum of Z, £hgp_ Cl(f-1(Int(W)))
is a strong continuum.

Proof. Since f is a continuous map of a compact
Hausdorff space onto a compact Hausdorff Space f is a closed
map. Therefore by Theorem 29 f-1(Int(W)) is connected. Since
f is continuous, f-1(Int(W)) is open. Thus, Cl(f-1(Int(W)))
is a strong continuum.

The following example shows that even if f is a closed
monotone map, f"1 of a strong continuum need not be a strong

continuum.

Example 3. Let S = 12, Z = {(x,y)‘k S x S l and

OSyS 1} and let A={(x,y)|0gxs!5 and OSyS 1}.
define f: S a Z by f(x,y) = (x,y) if % S x S l and
f(x,y) = (%,y) if 0 S x < %. Let ‘W = {(x,y)}(x - 3/4)2 +
(y - 1/4)2 S 1/4} U {(350)}0 S y S 1}. £‘1(w)= A U w,
Int(A) n W = ¢ and A n Int(W) = ¢.
Theorem 33. l££_ f be a monotone map of S gpgp. Z,
tree Y(f'1<A>) c Elmo).
Proof. Let A be a subset of z and suppose p e f-1(Y(A)),

then f(p) é Y(A). Hence there exist W 677((f(p)) such that

22

W m A = ¢. Therefore f-1(W) n f-1(A) = ¢. By Theorem 30
Cl(f-1(Int(W))) E 77((p), hence p 6 Y(A) and thus Y(f-1(A)) C
also».

Theorem 34. ESE f be a monotone map of S gpgg Z,
phgp f(Y(A)) C:Y(f(A)).

Theorem 35. lg£_ f be an open monotone map of S 9353
2. thee Y(f'1<A)) = f'liY(A>>.

Proof. By Theorem 33 Y(f-1(A))<: f-1(Y(A)), hence all
that needs to be shown is that f-1(Y(A))'C:Y(f-1(A)). Suppose
p é Y(f-1(A)), then there exist W E 771(p) such that
W n f-1(A) = ¢, hence f(W) n A = ¢. Since f is open
f(W) 6mm») and f(p) t Y(A). Thus p e Elmo) and
Flam) cw'lm).

Theorem 36. LEE f be an Open monotone map of S gppg
2, then Y(A) = f(Y(r'1(A)))-

Corollary 37. LEE f be an open monotone map of S
SEER. Z, EEEE

l. l£_ S is Y-additive then Z is Y-additive

 

2.

H

f S is Y-symmetric, then Z is Y-symmetric.

 

Proof. Let S be Y-symmetric and let A and B be

closed subsets of Z such that Y(A) n B ¢, then

Y(f'1(A>>.

f-1(Y(A)) n f-1(B) = ¢. Since f-1(Y(A))
Y(f-1(A)) n f-1(B) = ¢. Since S is Y-symmetric Y(f-1(B)) n
f-1(A) = ¢, thus Y(B) n A = ¢ and Z is Y-symmetric.

Let S be Y-additive and let A and B be closed
subsets of 2, then f-1(Y(A) u Y(B)) = Flam) u f-1,(Y(B)) =

Y(f'1(A>) u Y(f-1(B)) = Y(f'le) u 51(3)) = Y(f'ch u 3)) =

23

f-1(Y(A U 3)), thus Y(A) U Y(B) = Y(A U B) and Z is Y-additive.

Corollary 38. Let $1 and S

S1 X 82 is locally connected if and only if S1 and 32 are

2 be continua, then

locally connected.

Proof. Let P1 and P2 be the two projection maps,

then P1 and P2 are open monotone maps.

Let S1 X 82 be locally connected, then S and S

l 2

are Y-additive and hence it suffices to show that Y(a)

II
r-s

n)
a...)

for all a E 51' Let a 6 Sl,then Y(a) = P1(Y(P11(a))
P1(Y({a} X 82)) = P1({a} X 82) = a. Thus S1 and 82 are

locally connected.

Let 31 and 82 be locally connected. Let p 6 S1 X 82
and let A be any subset of S1 X 82 Such that p é Cl(A),
then there exist 01<: S1 and 02 C 82, both open, such that

p E 01 X 02 and (01 X 02) n A = ¢. Since S and S2 are

l
locally connected, there exist W1 6 7R(P1(p)) and W2 6 771(P2(p))
such that WICO1 and W2 C 02, hence p 6 W1 X W2 and
(W1 X W2) n Ai= ¢. Therefore p 4 Y(A). Thus by Theorem 4
S1 X 82 is locally connected.
Theorem 39. Let H gpd' K be closed subsets of S,

then the following are equivalent.

a. H n Y(K) = (b

b. There exists a finite collection C of strong

continua such that H is contained in the union

of the interiors of the elements of C and the

intersection of each element of C with K ii

empty.

24

c. There exist two closed subsets M Egg N _spgp
gig; H is a subset of the interior of M, K is
a subset of the interior of N _a_r_1_<_1_ M intersect
Y(N) is empty.

Proof. a implies b.

Let L be the Set of strong continua of S such that
Int(W) n H¥¢ and W0 K =9). Since HnY(K) =¢,

HC (U{Int(W)‘W E L}). Since H is compact, there exist
w1,...,wn such that Wi E L and HC (U[Int(wi)‘l S i S n}).
Let c ={wi\1 S i S n}.

b implies c.

Since S is normal there exist V such that V is
open H c: v c Cl(V) cu{1nt(W)\w e c} and there exist 11 such
that u is open KC 11 and Cl(U) n (U{w|w e 0}) = ¢. There-
fore C1(V) n Y(C1(U)) = 95. Let Cl(V) = M and Cl(U) = N.

c implies a.

K CN, therefore Y(K) C Y(N), H C M and M I] Y(N) = (I).
thus H n Y(K) = ¢.

The closed monotone image of a T-Sytrmetric compact
Hausdorff Space is a T-synmetric compact Hausdorff Space. This
final example shows that the closed monotone image of a Y-Symmetric
Space need not be Y-symmetric and that f(Y(A)) need not be
Y(f(A)).

Example 9. Let S be the closed line segments between
(-%,0) and (0,1) and the closed line segments between (Tl-’2)
and (0,3) unioned with [(x,% + Sine-HO S x S l} U

{(x,2% + sin(;]{")}0 < x S l} unioned with the closed line segment

25

between (0,0) and (0,3), then S is Y-synmetric and
Y((O,3)) = {(0,y)|2 s y s 3}.

Let 2 be the closed line segments between (-%,0)
and (0,1) unioned with those between (-%31) and (0,2)
for n 2 l unioned with [(x,% +-sin(§))}l < x S l} U
{(x,l% + sin(§))}0 < x S l} unioned with the closed line
segment between (0,0) and (0,2).

Let f((X.Y)) = (x,y) if y S 1. f((x,y)) = (0.1)
if 1 S y S 2 and f((x,y)) = (x,y-l) if 2 S y S 3.

Z is not Y-symmetric since Y((0,0)) = {(O,y)}0 S y S l}
and Y((O,2)) = {(0,y)|0 s y s 2}. f(Y(0,3)) # Y(f(0,3))
since f(Y((0,3))) = i({(0,y)\2 s y S 3}) = {(O,y)}l s y s 2}

and Y<f<<0,3))> = mm» = {(0.y)|o s y s 2}.

BIBLIOGRAPHY

BIBLIOGRAPHY

Davis, H.S., D.P. Stadtlander and P.M. Swingle, "Properties
of the set function Tn," Portugal Math. 21 (1962), 113-133.

 

Davis, H.S., "A note on connectedness Im Kleinen," Proceed-
ings of the American Mathematical Society, Vol. 19, No. 5,
October 1968, pp. 1237-1241.

Hocking, John G. and Gail S. Young, Topology, Addison-
Wesley.

Kuratowski, K., Topology Volume I, Academic Press.

 

Bourbasi, N., General Topology, Addison-Wesley.

 

Moore, R.L., Foundations of Point Set Theory, American
Mathematical Society, Volume XIII.

Fitzgerald, RHW., "The cartesian product of non-degenerate
compact continua is n-point aposyndetic," TOpology Conference,
Arizona State University, 1967, p. 324.

Dugundji, James, Topplogy, Allyn and Bacon.

Davis, H.S., To appear.

26

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