On asymmetry of multifunctions

12 (1991), 35-43

On asym m etry of m ultifunctions

W łodzim ierz Ślęzak

A m ultifunction F : X — » Y is a correspondence which assigns to each point in a set X at least one point in a set Y . If X and Y are topological spaces, F : X — » Y is said to be closed at a point xq G X

if for each point yo G F ( x 0) there exist two open neighbourhoods V (x 0) and U(yo) of th e points x 0 and y0, respectively, such th a t th e following im plication holds

x G V(a;o) = > F ( x 0) fi U( y0) = 0.

This notion was investigated e.g. in [11] under the nam e cofinal continu- ity. If F is cofinally continuous at each point x G X , it is briefly called cofinally continuous (or som etim es p-usc). F is cofinally continuous if and only if its graph

(1) G rF = {(®, y) : y € F ( x ) } C X x Y

is closed in the product space X X Y. It is easy to see th a t F is cofinally continuous if and only if its inverse m ultifunction : Y — * X , F -1 (y) = {x G X : y € ^ ( z ) } is cofinally continuous. If F : X — ■> Y is cofinally continuous at a point Xo, then the value F { x o) is closed in

Y. For com pact rangę spaces V , F : X — > Y is cofinally continuous if and only if F is upper sem icontinuous and has closed values. Let us recall, th a t a m ultifunction F : X — * Y is upper sem icontinuous if the inverse image

of each set D closed in Y , is closed in X . For m ore inform ations con- cerning to th e existing various kinds of continuity of m ultifunctions see [10], [11], [9], T he notion of cluster set for m ultifunctions is introduced in [14], [5] and [8] as follows. We say th a t y is an elem ent of th e cluster set of F : X — » Y at x, denoted C ( F , x ), if there exists a net Xd G X , d G D and a net yj G F(xd) such th a t Xd is M S-convergent to x and yd is M S-convergent to y. D is here an apropriate directed set. Thus th e cluster set m ay be tre a ted as a form alization of th e set of ”lim it po in ts” of our m ultifunction F.

D enoting by CL th e closure operato r in Y , we have C ( F , x ) = P){C L F (f7) : U G N ( x ) } ,

where N ( x ) denotes th e filterbase of neighbourhoods at x, and the im age F( U) is defined for m ultifunctions as follows

H U ) = U { C ( x ) : X e U ) .

In [5] th e following theorem is essentially proved.

t h e o r e m 0 [5]. The following conditions are equivalent fo r m ultifunc­ tion F : X — * Y

(i) F has closed graph (1) i.e. F is cofinally continuous (11) F( x ) = Pr y ({*} x Y n C L (G rF )), x G X

(iii) C ( F , x ) = F( x ) , x G X .

An inspection of the proof gives also a local version of th e above th e ­ orem: F is cofinally continuous at x if and only if F( x ) = C ( F, x ) . In a case if F is single valued the cofinal continuity reduces to usual con­ tin u ity provided th e rangę space is com pact and then we obtain from theorem 0 th e result of Weston [13]. T he reader can easily construct an exam ple where c a rd F (x ) = 1, i.e. F( x ) — { f { x ) } , C ( f , x ) is a singleton yet / is not continuous using a non-com pact rangę space Y. At present we define a relation on topologies analogous to one defined by Ulysses H u n ter in [4]. Applying cluster set techniques we obtain some inform ation on the set of cofinal discontinuities of an arb itrary

m ultifunction. This seems to be of some interest in connection with th e result of [6], where th e conditions on Borel m easurability of m u lti­ functions of two variables are form ulated and also in connection w ith th e result of [1], where in term s of cluster sets J. Ceder characterised m ultifunctions possessing a selector with the D arboux property.

A collection M of subsets of X is called a cr-ideal of sets in X if (a ) A G M and B C A implies th a t B G M ,

(b ) A n G M, n = 1 , 2 , 3 , . . . , implies th a t (J^Li A„ G M , (c) X $ M.

Two topologies T and S on A' are said to be related modulo M , denoted T rei S m od M , if for any subset A of X , the 7 '— and S closure of A differ by a set in M

CL t( A) A CL5 (A) G M, A A B = (A \ B ) U ( B \ A ) .

T h e o r e m 1 Let M be a cr-ideal o f subsets o f a bitopological space (X , T , S ) and T rei S m od M . I f F : X — * Y is an arbitrary m ultifunction fro m the space X into the second countable space Y , then

Ct{ F,x) = Cs { F , x ) fo r euery x G X \ A fo r some A G M P roof. Let {[/„ : n = 1 ,2 , 3 , . . . } be a countable open basis for Y . Observe th a t A sy m F = {x G X : Ct{ F,x) ^ Cs { F, x ) } = E U D, where E = { x e X : Ct( F ,x) £ C s( F ,x)} and D = { x e X : Cs( F ,x) £ C t( F ,x)}. To show th a t E G M , p u t for n = 1 , 2 , . . .

where th e big inverse image is defined by form uła (2). N ote th a t in com pliance w ith th e assum ption th a t T rei S m od M , each E n be- longs to M and hence, by v irtu e of (b) also U^Li E n is in M. We claim th a t E is included in this union U^Li E n. Indeed, if x is in E then there exists an y in Y such th a t for every V G N( y ) and for every G G N ( x )

F ( G)

n

0

CL

b u t F ~ ( N ( y ) ) does not accum ulate at x w ith respect to S, th a t means th ere is a V\ in N ( y ) and G\ in N s ( x ) such th a t

F ( G i ) n V i =

0

Since {Un : n = 1 ,2 ,3 ,...} is a basis for Y , th ere exists some positive integer k such th a t y G Uk ę V\. We conclude th a t x is in Ek and hence we have th e claim. Thus E G M and a sim ilar argum ent shows th a t D G M , com pleting the proof.

R e m a r k 1 In case of single valued functions this theorem reduces to a p a rt of theorem 1 on p.78 in [12]. N ote th a t Świątkowski wrongly assum ed the separability of Y instead of second axiom of countability in his theorem .

T h e o r e m 2 Let F : X — > Y be an arbitrary m ultifunction fro m X into the second countable H ausdorjf space Y . Let M be a a -id ea ł o f subsets o f X with T and S two topologi.es on X related modulo M . (a ) I f F is cofinally continuous with respect to T , then the set o f cofinal

discontinuities o f F with respect to S is an element o f M . (b ) I f F : X — > Y is either cofinally continuous with respect to S

or with respect to T at. euery point o f X , then the set o f cofinal discontinuities with respect to the intersection T fi S is an element o f M .

P r o o f . By theorem 0 we have F( x ) = Ct{ F, x)for all x G X . A pplying theorem 1 we find a set A G M such th a t C s ( F , x ) = Ct{ F ,x) for x G X \ A. Therefore F( x ) = Cs { F, x ) for x G X \ A and, taking into account once again th e local version of theorem 0 we deduce th a t F is

cofinally continuous w ith respect to S at all points x G X \ A. To prove p a rt (b) of Theorem 2, first observe th a t

Ct{ F,x) n Cs ( F , x ) C CTn s { F ,x ) C CT ( F, x ) U Cs ( F, x) .

Now, th ere exists an subset A belonging to M such th a t for x G A \ A we have th a t Ct( F ,x) — Cs ( F, x ) . Since our m ultifunction F is cofinally continuous eith er w ith respect to T or w ith respect to S, we have th a t

C t(F , x) = CS (F, x) = F{ x ) C Y iff x € X \ A.

Hence, if x G X \ A we have C rns{F , x) = F( x ) and conseąuently F is cofinally continuous at x G X \ A with respect to intersection of topologies T fi S. We conclude th a t the set of cofinal discontinuities of F w ith respect to T Pi S is a subsęt of A and hence an elem ent of M . The proof is complete.

R e m a r k 2 This theorem even in single-valued case is stronger th an th e corresponding one in [3] (th. 3.5). Namely H am lett dealt w ith the topology generated by th e union T U S of original topologies instead if its intersection.

In case w here Y is com pact we obtain the following corollary.

C o r o lla r y 1 Let F : X — » Y be an upper sem icontinuous m ultifunc­ tions with compact rangę Y and closed ualues. IJ S is another topology on X which is related to the initial topology T modulo M , then the set o f points at which F fails to be upper sem icontinuous with respect to S is an elem ent o f M . Moreouer i f at each point x G X , F is either upper sem icontinuous with respect to T or upper sem icontinuous with respect to S , then it is upper sem icontinuous with respect to both topologies sim ultaneously, except o f a set o f points belonging to M .

An im p o rtan t conseąuence of Theorem 2 in case where X is the real line is th e following:

T h e o r e m 3 Let F : X — > Y be an arbitrary m ultifunction fro m the real line into a second countable H ausdorff space. I f F is either cofinally continuous fro m the right or cofinally continuous fro m the left at euery

real number x G X , then F has at m ost countably m any points, at which it fails to be cofinally continuous with respect to the Euclidean topology on X .

P r o o f . Let M be th e er-ideal of countable sets of th e real line X . If R is th e topology on X generated by {[x, x + r) : x G X , r > 0} and L is th e topology on X generated by {(a: — r,x ] : x G X , r > 0} then it is easy to show th a t L is related to R m odulo M (see e.g. H unter [4], E xam ple 1). Observe th a t L fi R is the usual Euclidean topology. It suffices to apply Theorem 2 to obtain th e thesis.

In case when Y is com pact and F has closed values, a sim ilar th e­ orem is valid for upper sem icontinuity in place of cofinal continuity. If Q is a subset of then we can define the cluster set of F : X — > Y at x G X relative to Q as follows (cf. [8], [5]):

C ( F , x , Q ) = f ) C L F ( U n Q ) ,

w here th e intersection is taken over all neighbourhoods U G N ( x ) of x in X w ith F { U fi Q) = U{E(2) : t G U H Q ] .

From this definition it is elear th a t C { F , x , Q ) is ałways closed ( bu t possibly em pty ) subset of Y . In the theory of m ultivalued functions of complex variable esspecially im p o rtan t are radial, angular and curvi- linear cluster sets obtained from (15) by suitably choosen Q. In order to generalize th e Theorem 3 to higher dimensions we m ust define a no- tion of a point of asym m etry of a m ultifunction F : X — » Y where X stands for an n-dim ensional Euclidean space. Let in this space, besides th e n a tu ra l topology, another topology T be distinguished.

We will denote by CL^A , D eryH etc. th e closure of th e set A C X , th e set if its accum ulation points, etc. w ith respect to th e topology T. W hen th e topology T coincides w ith th e n a tu ra l topology, we om it th isv additional notation.

A point i G l will be called a T -assy m m etry point of a m ultifunction F : X — > y if th ere exists an (n — l)-d im en sio n al hyperplane H passing through x such th a t

Ct( F ,x, X +) Ć Ct( F ,x, X - ) ,

w here X + and X ~ denote th e halfspaces being the com ponents of the set X \ H. O bserve th a t Theorem 3 on countability of th e set of

asym-m etry points of a asym-m ultifunction of one real variable cannot be car- ried over to m ultifunctions of m any variables, as the following exam ple shows

(17) R ’ l { x , » ) ^ F { x , y ) = { [ ^

However, in this case the cr-ideal of countable sets m ay be replaced by th e u -id e a l of m eager sets

T h e o r e m 4 Let F : X — > Y be a m ultifunction defined on an n -d i- m ensional Euclidean space X and takmg his uahies in a second count­ able space Y . The set o f assym etry points o f F is o f the first category. P r o o f . A cting as in th e proof of Theorem 2 and 1 it suffices to prove th a t the sets

A sym B = D er(B fi A + ) A Der(Z? H X - ),

for B = F~ ( Un), n = 1 , 2 , . . . are nowhere dense. Namely let B C X and let K be an a rb itra ry bali in th e space X . We will show th a t th ere exists a bali K i C K disjoint w ith the set A sym B. F irst, if K is disjoint w ith A sym B it suffices to pu t AT — K . So let x £ K fi AsymjB. C onseąuently there exists a decom position X = X + U H UA'~ corresponding to th e point x and such th a t x £ A sym B. W ith o u t loss of generality we m ay assum e th a t x 0 Der(Z? D A'+ ). Then there exists a bali K 0 centered at x such th a t KqC \X+ C\B — 0. Now let K \ be a bali

contained in I<0 D X +. The set AT fi B C K 0 D X + (T B m ust be em pty which implies th a t th e bali AT does not contain any accum ulation point of th e set B and conseąuently any of its asym m etry points. T h a t ends th e proof.

R -e m ark 3 This proof is alm ost idenitcal with the proof of Theorem 4 in [12], only th e m eaning of the set B is different. Note th a t a form uła (7) in T heorem 3 in cited Świątkowski’s paper concerns th e points of T -sy m m etry , b u t not T -asym m etry, as it is errorously statecl.

R epeating th e proofs of Theorem 5 and 6 from [12] w ith th e obvious changes we can ob tain a characterization of T -asy m m etry points where

T is density topology, w ith respect to the ordinary differentiation basis. Such points will be called ordinary approxim ative asym m etry points.

In fact, we have

T h e o r e m 5 The ordinary approximative asym m etry points o f a mul- tifunction F : X — > Y defined on a fin ite dim ensional Euclidean space X fo rm a set o f the first category. Moreouer, the Lehesgue measure o f the set o f ordinary approximative asym m etry points o f F is equal to zero.

T he cr-porosity of the set of approxim ative sym m etry points, in the spirit of [15], as well as asym m etry points w ith respect to another fine topologies (H ashim oto topology e.g.) will be investigated in a later paper.

R e f e r e n c e s

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[11] S troth er W . L., C ontinuity fo r multiualued fu n ctio n s and some application to topology, Tulane University 1951.

[12] Świątkowski S., On some generalization oj the notion o f asym m e- l ry o f functions, Coli. M ath. XVII fasc.l (1967), 77-91.

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