Connectedness oî the hyperspace of closed connected subsets

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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRAOE MATEMATYCZNE X X (1978)

A. Y. W.

Lau

and 0. H.

Voas

(Denton)

Connectedness oî the hyperspace of closed connected subsets

Abstract. Let A be a connected Hausdorff space and C( X) be its collection of closed connected subsets with the Vietoris topology. An example is given to show that 0 ( X) is not necessarily connected. A sufficient condition is given to obtain connected 0{ X) . This condition implies that the “explosion point” example of Knaster and Kuratowski has connected C(X).

I t is known th a t if X is a connected T x space, then 2X (the hyper

space of non-empty closed subsets of X) is connected. I t is also known th a t if X is a continuum ( = compact connected T z), then G(X) is a con

tinuum.

Let X be a space and C(X) be the collection of all non-empty closed connected subsets of X . The Vietoris topology on C(X) is generated by sets of the form < VX1 . . . , F n>, where each V{ is open in X and A e <F1? ...

..., Fn> iff A e G{X), ArGVi 0 for each i and A £ {F*

1

i =

1

, ..., n}.

Denote the closure of A by A. We will discuss when C(X) is connected* if X isconnected. A counterexample will be given and some sufficient conditions are provided to obtain connected C{X).

A space X has property (1) provided there exists p e X such th a t if p e V and V is open in X, then there is a non-degenerate K e G(X) such th at K s V.

Theorem 1

. I f X is a non-degenerate connected T z space and G(X) is connected, then X has property (1).

P ro o f. Suppose for each p e l , there exists an open set Vp con

taining p such th a t ( V p) = (Vp) n l , where X is the set of all singletons in X. Then X = U {(F^) jp e X } which is open in G(X). But X is closed since X is T z . This contradicts €{X) being connected.

Example

A. Let X be Bing’s countable connected Hausdorff space in [1]. Then X does not have property (1); hence C(X) is no t connected.

Example

B. Let p be a point on the ж-axis in Bing’s example and

adjoin an arc I to X with one endpoint of I equal to p. Then X u G ( I )

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394 A. Y. W. Lau and С. H. Yoas

is closed and open in <7(XuI). Furthermore, X u I has property (1) at each point of I.

A space X has property (

2

) provided th a t if К is a non-degenerate proper closed connected subset of X and K £ V open in X, then there exists H

^e

C(X) such th a t K £ H £ V and К Ф II.

Theorem 2.

I f X is connected T z and has properties

(1)

and

(2),

then C{X) is connected.

P ro o f. Let C(X) be separated by °U and У . Since X is connected, one can assume X £ У. Let p be the point given by property (

1

). Then fp)

е

У implies th a t there is an open set V such th a t {pj e <F> £ У.

Choose a non-degenerate К

e

C(X) and K £ V. Let ^ = {H

е

У \ К £ H}

and A < В iff A £ B. Choose a maximal chain J t in # and put M —

= (U -* )* e C (X ).

Let M e <T'rx, ..., Vf). Then for each i, there exists М {

^e

Л and M i C\Yi Ф 0 . Since {J1J is a chain, then there is a largest Mj which meet each Ту, i.e., M

^e

I = У . Property (2) and the openness of У imply* M = X.

Let P

^е^<Ш.

Since X £ У , then P is non-degenerate. Do the same con

struction for P in Щ. One obtains X

^e

% which is a contradiction.

Corollary 1

. I f X is Т г and X = \ J K a and П ^ а # 0 , where each K a is a continuum, then C(X) is connected.

P ro o f. Let p

e f ]

K a and assume X is non-degenerate. Then p

e

Kp for some non-degenerate continuum K p. We will show th a t X has prop

erties (

1

) and (2). Let p

^e ^V

open in X. Choose 77 open in K p such th a t p

^e

77 £ 77 £ V nKp. Then the component of p in 77* is a non-degen*

erate continuum contained in V.

Let О be a non-degenerate proper closed connected subset of X and C £ 77 open in X. Assume p

e

C. Since <7 Ф X, there exists K a which is not contained in C. Let C n K a £ V £ V £ U n K a, where V is open* in K a. The component В of p in V meets the boundary of V relative* to K a. Hence C uD is the required closed connected subset contained in 77.

Suppose p фС. Choose any q E C n K a and do the same construction as in the previous paragraph.

Corollary 2

. I f X is locally connected T3 and connected, then <7(X) is connected.

P ro o f. To prove properties (1) and (2), let G

e

C(X), and С Ф X and C £ W open in X. Since C is not open, there exists

qE(

7 which is not an interior point of (7. Choose a connected open set 77 such th a t

q ^e

77 £ 77 £ W. Then С и 77* is the required closed connected set.*

Example

C. The converse of Theorem 2 is false. If n is a positive

integer, let Ln — {(t, tfn) JO < t < 1} and L = \ J B n. P ut 77 = {(a?,0)|

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Connectedness of the hyperspace 395

0 < ж < 1 and x Ф 1/2} and X = L u TJ. Since M = {K e G(X) | (0, 0) e K}

is connected (same proof as Theorem 2) and each C{Ln) meets M at {(0, 0)}, then l u ( ( J C ( i j ) is connected. I t suffices to show it is dense in C(X).

Let К e G(X) and (

0

,

0

) ф K. Then К я L n or К я TJ. Assume К я TJ.

Then К = U K j , where {Kj} is an increasing sequence of compact connected subsets of TJ. If K e <F1? Vn), then some e <JVX, ..., Fn>. There exists A e G { L n) snch th a t i e <F3, 7 B>. Note th a t property (2) fails a t К = {(x,

0

) j-|< x <

1

}.

Example

D. The “explosion point” example of Knaster and Kura- towski (see [3], p. 22] will be shown to have properties (1) and (2). Let G x {

0

} be the canonical Cantor set on the ж-axis and Y the union of all line segments L(x) from (ж, 0) to a = (J, £). Let G = PuQ , where P consists of endpoints and Q its complement in G. Then the explosion point example X is the set of all (z,t) e L{x), where t is rational iff ж e P and t is irrational iff x eQ. Note th a t A and В (A) mean closure and* boundary of A in У (not in X).

I t is easy to see th a t property (1) is satisfied a t a (and nowhere else).

The following two lemmas show X has property (

2

).

Lemma 1

. I f H e 2X, a e H, H я TJ which is open in Y, then F — {x e G\L(x)nHnB(TJ) — 0} is of first category in G.*

P ro o f. Let Iг = 1 , 2 , . . . } be all the rationals in [

0

, \) and G{

= {{b, r{) e H nB( U)\ (b, rf) e L(x) for some x e Q} .*

Let ж e F n Q and g e L{x)nH nB(TJ). Since g $ TJ, then g фН. But* H — H n X yields g ф X. The y-coordinato of g is rational which gives* us g e G{ for some i.

Note th a t G я H* n B ( 17) n {(b, rt) | (b, rf) e Y}. But if qeG, then q ф X since H = X n H . So q e L(x) for some x eQ. Let F t be the image* of the (stereographic) projection of G down the line segments into the* Cantor set. F t is nowhere dense since Fi n P = 0 . Moreover, F я ( U Ff juP.

Lemma 2

. I f H e G( X) and H is non-degenerate and proper, H я TJ open in Y, then there exists L e G(X) and H я L я TJ and Н Ф L.

P r o o f. We know a e H and by the previous lemma, one can get

® s G such th a t L(x)nHnB(TJ) = 0 . Note th a t L{x)nH* is an arc.*

In fact, we can choose ж such th a t {а] Ф L{x)nH Ф L(x). Since a* e L ( x ) n H which is connected and (L(x)nH)nB(TJ) = 0 , then L{ x ) n пЯ s TJ. Choose {e, t) e (L{x)nX)\B. * such th a t {(/, s) e L(x) |s ^ t}*

£ TJ. Choose M e G( X) such th a t (e, t) e M and M я U. Now L — H\

j

M is the appropriate connected set in X since a e M and (e,t) e M \ H .

R e m a rk . In contrast to hyperspaces of continua: (a) there are maximal chains in G(X) which are not connected, (b) {a} is a cutpoint of G{X).

11 — Roczniki PTM Ртасе Mat. XX.2

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396 A. Y. W. Lau and С. H. Voas

R e f e r e n c e s

[1] E. H. B in g , A connected countable Hausdorff space, Proo. Amer. Math. Soo.

4 (1953), p. 474.

[2] Jack T. G o o d y k o o n tz , Jr., Connectedness im kleinen and local connectedness in 2X and 0( X) , Pacific J. Math. 53, 2 (1974), p. 387-397.

[3] W. H u r e w ic z and H. W a llm a n , Dimension theory, Princeton University Press, 1948.

[4] K. K u r a to w s k i, Topology, Yol. 2, Academic Press, 1968.

[5] J. T. R o g e r s, Jr., TTie cone-hyperspace property, Can. J. Math. 24, 2 (1972), p. 279-285.

[6] A. D. W a lla c e , A theorem on acyclicity, Bull. Amer. Math. Soc. 67 (1961), p. 123-124.

DEPARTMENT OF MATHEMATICS NORTH TEXAS STATE UNIVERSITY DENTON, TEXAS

Connectedness oî the hyperspace of closed connected subsets

A. Y. W.

and 0. H.

(Denton)

Connectedness oî the hyperspace of closed connected subsets

I t is known th a t if X is a connected T x space, then 2X (the hyper­

space of non-empty closed subsets of X) is connected. I t is also known th a t if X is a continuum ( = compact connected T z), then G(X) is a con­

tinuum.

Let X be a space and C(X) be the collection of all non-empty closed connected subsets of X . The Vietoris topology on C(X) is generated by sets of the form < VX1 . . . , F n>, where each V{ is open in X and A e <F1? ...

..., Fn> iff A e G{X), ArGVi 0 for each i and A £ {F*

i =

, ..., n}.

Denote the closure of A by A*. We will discuss when C(X) is connected if X isconnected. A counterexample will be given and some sufficient conditions are provided to obtain connected C{X).

A space X has property (1) provided there exists p e X such th a t if p e V and V is open in X, then there is a non-degenerate K e G(X) such th at K s V.

. I f X is a non-degenerate connected T z space and G(X) is connected, then X has property (1).

P ro o f. Suppose for each p e l , there exists an open set Vp con­

taining p such th a t ( V p) = (Vp) n l , where X is the set of all singletons in X. Then X = U {(F^) jp e X } which is open in G(X). But X is closed since X is T z . This contradicts €{X) being connected.

A. Let X be Bing’s countable connected Hausdorff space in [1]. Then X does not have property (1); hence C(X) is no t connected.

B. Let p be a point on the ж-axis in Bing’s example and

adjoin an arc I to X with one endpoint of I equal to p. Then X u G ( I )

is closed and open in <7(XuI). Furthermore, X u I has property (1) at each point of I.

A space X has property (

) provided th a t if К is a non-degenerate proper closed connected subset of X and K £ V open in X, then there exists H

C(X) such th a t K £ H £ V and К Ф II.

I f X is connected T z and has properties

and

then C{X) is connected.

P ro o f. Let C(X) be separated by °U and У . Since X is connected, one can assume X £ У. Let p be the point given by property (

). Then fp)

У implies th a t there is an open set V such th a t {pj e <F> £ У.

Choose a non-degenerate К

C(X) and K £ V. Let ^ = {H

У \ К £ H}

and A < В iff A £ B. Choose a maximal chain J t in # and put M —

= (U -* )* e C (X ).

Let M e <T'rx, ..., Vf). Then for each i, there exists М {

Л and M i C\Yi Ф 0 . Since {J1J is a chain, then there is a largest Mj which meet each Ту, i.e., M

I * = У . Property (2) and the openness of У imply M = X.

Let P

Since X £ У , then P is non-degenerate. Do the same con­

struction for P in Щ. One obtains X

% which is a contradiction.

. I f X is Т г and X = \ J K a and П ^ а # 0 , where each K a is a continuum, then C(X) is connected.

P ro o f. Let p

K a and assume X is non-degenerate. Then p

Kp for some non-degenerate continuum K p. We will show th a t X has prop­

erties (

) and (2). Let p

open in X. Choose 77 open in K p such th a t p

77 £ 77* £ V nKp. Then the component of p in 77* is a non-degen­

erate continuum contained in V.

Let О be a non-degenerate proper closed connected subset of X and C £ 77 open in X. Assume p

C. Since <7 Ф X, there exists K a which is not contained in C. Let C n K a £ V £ V* £ U n K a, where V is open in K a. The component В of p in V* meets the boundary of V relative to K a. Hence C uD is the required closed connected subset contained in 77.

Suppose p фС. Choose any q E C n K a and do the same construction as in the previous paragraph.

. I f X is locally connected T3 and connected, then <7(X) is connected.

P ro o f. To prove properties (1) and (2), let G

C(X), and С Ф X and C £ W open in X. Since C is not open, there exists

7 which is not an interior point of (7. Choose a connected open set 77 such th a t

77 £ 77* £ W. Then С и 77* is the required closed connected set.

C. The converse of Theorem 2 is false. If n is a positive

integer, let Ln — {(t, tfn) JO < t < 1} and L = \ J B n. P ut 77 = {(a?,0)|

0 < ж < 1 and x Ф 1/2} and X = L u TJ. Since M = {K e G(X) | (0, 0) e K}

is connected (same proof as Theorem 2) and each C{Ln) meets M at {(0, 0)}, then l u ( ( J C ( i j ) is connected. I t suffices to show it is dense in C(X).

Let К e G(X) and (

,

) ф K. Then К я L n or К я TJ. Assume К я TJ.

Then К = U K j , where {Kj} is an increasing sequence of compact connected subsets of TJ. If K e <F1? Vn), then some e <JVX, ..., Fn>. There exists A e G { L n) snch th a t i e <F3, 7 B>. Note th a t property (2) fails a t К = {(x,

) j-|< x <

}.

D. The “explosion point” example of Knaster and Kura- towski (see [3], p. 22] will be shown to have properties (1) and (2). Let G x {

I t is easy to see th a t property (1) is satisfied a t a (and nowhere else).

The following two lemmas show X has property (

).

. I f H e 2X, a e H, H я TJ which is open in Y, then F — {x e G\L(x)nH*nB(TJ) — 0} is of first category in G.

P ro o f. Let Iг = 1 , 2 , . . . } be all the rationals in [

, \) and G{

= {{b, r{) e H* nB( U)\ (b, rf) e L(x) for some x e Q} .

Let ж e F n Q and g e L{x)nH* nB(TJ). Since g $ TJ, then g фН. But H — H* n X yields g ф X. The y-coordinato of g is rational which gives us g e G{ for some i.

. I f H e G( X) and H is non-degenerate and proper, H я TJ open in Y, then there exists L e G(X) and H я L я TJ and Н Ф L.

P r o o f. We know a e H and by the previous lemma, one can get

I t is known th a t if X is a connected T x space, then 2X (the hyper

space of non-empty closed subsets of X) is connected. I t is also known th a t if X is a continuum ( = compact connected T z), then G(X) is a con

Denote the closure of A by A. We will discuss when C(X) is connected* if X isconnected. A counterexample will be given and some sufficient conditions are provided to obtain connected C{X).

P ro o f. Suppose for each p e l , there exists an open set Vp con

I = У . Property (2) and the openness of У imply* M = X.

Since X £ У , then P is non-degenerate. Do the same con

Kp for some non-degenerate continuum K p. We will show th a t X has prop

77 £ 77 £ V nKp. Then the component of p in 77* is a non-degen*

C. Since <7 Ф X, there exists K a which is not contained in C. Let C n K a £ V £ V £ U n K a, where V is open* in K a. The component В of p in V meets the boundary of V relative* to K a. Hence C uD is the required closed connected subset contained in 77.

77 £ 77 £ W. Then С и 77* is the required closed connected set.*

. I f H e 2X, a e H, H я TJ which is open in Y, then F — {x e G\L(x)nHnB(TJ) — 0} is of first category in G.*

= {{b, r{) e H nB( U)\ (b, rf) e L(x) for some x e Q} .*

Let ж e F n Q and g e L{x)nH nB(TJ). Since g $ TJ, then g фН. But* H — H n X yields g ф X. The y-coordinato of g is rational which gives* us g e G{ for some i.

® s G such th a t L(x)nHnB(TJ) = 0 . Note th a t L{x)nH* is an arc.*

In fact, we can choose ж such th a t {а] Ф L{x)nH Ф L(x). Since a* e L ( x ) n H which is connected and (L(x)nH)nB(TJ) = 0 , then L{ x ) n пЯ s TJ. Choose {e, t) e (L{x)nX)\B. * such th a t {(/, s) e L(x) |s ^ t}*