RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo XXXIX (1990), pp. 427-435

C O N T I N U O U S S E L E C T I O N S F R O M T O P O L O G I C A L

TO M E T R I C SPACES

IVAN KUPKA

Soient X un espace topologique et Y un espace m6trique. Soit /? : X ~ Y tree multifoncfion continue au sens de Hausdorff.

Darts l'arficle prdsent nous d6montrons une condition suffisante pour l'existence de s61ecteur continu de F.

1. Introduction.

The study of continuous selections begins with the papers of

Michael and is an active area of research in general topology. Michael

[6] proved that any lower semicontinuous multifunction F : X - . - , Y

with closed convex values admits a continuous selection, where X is

a paracompact topological space and Y is a Banach space.

For continuous multifunctions which are defined on the segment

[0,1] with compact values in Euclidean space E n the existence of

a continuous selection was proved by Filippov [2] for multifunctions

satisfying the Lipschitz condition, by Hermes [4] for multifunctions

with bounded variation. Several authors obtained a continuous selection

theorem for a multifunction with non convex values in a Banach space

of functions (e.g. Fryszkowski [3]).

428 IVAN Zt~ZA

The present paper answers the following remark of T. Parthasarathy

in [7] (p. 87): <<It could be nice if one could prove a selection

theorem when Y is an arbitrary metric space (X being topological) -

of course one has to have strong assumptions on the map F as well

as the sets F(z)>~.

Carbone in [1] proved the existence of a continuous selection for

a 1.s.c. multifunction F : X ~ Y with exactly n values (or exactly

infinite many values) where X is a connected sigma-compact metric

space. But to answer the question of [7] completely we need X to be

an arbitrary topological space.

This paper gives a selection theorem for such an X.

2. preliminaries.

Let X and Y be two sets; Let P(Y) denote the set of all

nonempty subsets of Y. A multifunction from X to Y is a function

F from X to P(y). We write F : X ~ Y .

A selection for a multifunction F : X ~ Y is a function f : X ~ Y

such that for each x in X f(x) is an element of F(x).

Let X and Y be topological spaces. A multifunction F from X

to Y is called lower semicontinuous (I.s.c.) if for each open subset G

of Y the set F-(G)= { x ; F ( x ) A G#0} is an open subset of X.

We are going to define the notion of Hausdorff continuity of a

multifunction from a topological space X to a metric space Y. First

let us define the Hausdorff distance H between two subsets C and K of Y to be

H(C, K) = inf{s > 0; B~(C) D K and B~(K) D C}

where BE(C) (resp. B~(K)) denotes the union of all open E-balls

whose centers mn over C (resp. over K).

We note that distance so defined might not yield a finite number.

But if we restrict our distance function to the closed and bounded

subsets of Y, we do obtain a metric, called the Hausdorff metric ([5]).

CONTINUOUS sm_~-noNs FROM TOeOt~CAL TO MmmC SPACES 429

Let F be a multffunction from a topological space X to a metric

space Y. Then F is said to be Hausdorff continuous at x E X provided

that for each e > 0 there is an open neighborhood U of x such that

if t E U then H ( F ( x ) , F( t ) ) < e. If F is Hausdorff continuous at each

point of X , then F is said to be Hausdorff continuous.

3. The main result.

Before proving Theorem 1 we need the following technical lemma.

LEMMA 1. Let X be a topological space. Let (Y, d) be a metric

space. Let F : X ~ Y be a Hausdorf f continuous multifunction such,

that for each x E X the set F (x ) is not a singleton. Let v : Y ~ R

be a uniformly continuous function. I f we define a function c : X ---, Y

as follows:

c(z) = inf{Iv(a) - v(b)l; a, b E F(x) , acb}

then the function c is continuous.

Proof. Let x E X. Let e > 0 be arbitrary.

Using the uniform continuity of v we

r > 0 such that

obtain that there exists

Va, b E Y such that d(a, b) < r e holds. I v ( a ) - v(b)l <

Let Ux be a neighborhood of x such that

(i) for each t in Ux H(F( t ) , F(x) ) < r

where H denotes the Hausdorff distance induced by d.

Let t E Ux.

(1) To prove c ( t ) < c ( x ) + e let z, s E F ( x ) such that

E IIv(s) - v ( z ) l - c(x) l <

430 WAs Ktn'KA

8 ! By (i) there exist z', s F ( t ) such that d(z, z') < r and d(s, 8') < r. E E

Hence Iv ( s ) -v ( s ' ) [ < ~ and [v (z ) -v (z ' ) [ < ~ so

(iii) IIv(z')- ~(8')1- I v ( z ) - v(~)ll <

holds.

Therefore by (ii) and (iii) IIv(z')-v(~')l-c(z)l < ~ so Iv(z')-v(~')l < c (x )+e and since c ( t ) < [ v ( z ' ) - v(s')[, c( t )< c (x )+e holds.

(2) To prove c ( z ) < c ( t ) + e let p, q be elements of F(t) such that

E IIv(p) - v ( q ) l - c(t)l < ~.

By (i) there exist p' ,q' from F(x) such that d(p ,p ' )< r and

d(q, qt) < r so

and

E Iv(p) - v(p') < -~

Iv (q ) - v(q')l < ~.

The rest of the proof of (2) is same as in (1) and is left to

the reader. So we have Vt E Ux I c ( x ) - c(t)l < ~ and the proof is completed.

THEOREM 1. Let X be a topological space, let (Y, d) be a metric

space. Let F : X ~ Y be a Hausdorff continuous multifunction such,

that for each x E X the set F(x) is not a singleton. Suppose that

there exists a uniformly continuous function v : Y ~ R such that

(i) For each x in X the set v(F(x)) is bounded below.

(ii) For each x in X

inf Iv(a) - v(b)l; a, b E F(x) and aCb} > 0 holds.

Then F has a continuous selection.

Proof. first let us define a function c : X ~ R as follows:

c(x) = inf{Iv(a) - v(b)]; a, b E F(x) and acb}.

CONTINUOUS SELECTIONS FROM TOPOLOGICAL TO METRIC SPACES 431

By Lemma 1 c is continuous and by (ii) c ( z ) > 0 for each z in

X.

It is easy to verify, using (i) and (ii) that for each z E X the set

v(F(z)) has the least element and that there exists exactly one point

Y E F (z ) such that v(y) = minv(F(z)) .

Let us denote the minimizing element Y as f (z) . Then f is a

function from X into Y. Obviously f is a selection for F . We will

prove that f is continuous.

To prove this let z E X and e > 0 . We will show that there

is an open set Uz in X such that z E Uz and for each t 6 U z

d(f(t); f (z ) ) < e holds.

First take r > 0 such that c(z )> r. Next let h > 0 be such that

(v) h < e and for all a ,z E V such that d ( s , z ) < h the inequality

Iv(s) - v(z) I < r holds.

Since the function c is continuous at z and F is Hausdorff

continuous at z there exists an open neighborhood Ox of x such that

(a) for each t E Oz c(t) > r and

(b) for each t E Oz H(F(z) , F(t)) < h hold.

Suppose that, contrary to what we wish to prove,

(c) There exists s E Ox such that d(f(s), f(:c)) >_ e

Using (b) and the fact that f is a selection for F we obtain

There exists m E F(s) such that d(f(x), m ) < h

There exists n E F(x) such that d(f(s), n) < h

It follows from (c) that m:~f(a) and n~f(z) . Now using (v) it

follows that

(bl) (b2)

(vl) I v ( m ) - v(f (z ) ) l < r

(v2) Iv(n) - v(f(a))l < r.

It is clear now (using the definition of f and c,

assumption (ii)) that v ( f ( s ) )+r < v(m) and (vl) implies

v(f (s ) ) < v(f(z)).

(a) and the

432 IVAN ~:tn'KA

Analogicaly v ( f ( z ) ) + r < v(n) holds and (v2) implies

v ( f ( z ) ) < v( f(s)) .

This is a contradiction, t lence for each s C Oz d(f(s) , f ( z ) ) < e

holds and we are done.

Remark If F is a multifunction fulfilling the assumptions (i)

and (ii) of Theorem 1 then for each z in X the set F ( z ) has no

accumulation point.

We now present an exemple relevant to Theorem 1.

EXAMPLE Let A = [0, 2], B = [0,4] be closed real intervals. Let

X be a circle (a quotient space) obtained from A by identifying the

points 0 and 2. Let Y be a circle obtained from B by identifying the

points 0 and 4. Let us denote the identified points 0 and 2 in A by s

and 0 and 4 in B by z.

We now define F : X --, Y as follows:

F(s) = {2, z}

F(t ) = {t, 2 + t} for each t E (0, 2).

It is easy to see that F is 1.s.c. at each t in X. Thus by Lemma

2 below F is Hausdorff continuous.

But there is no continous selection f : X ~ Y for F . If a

continuous function g : X - . Y were a selection for F then it would

be defined by g ( t ) = t for all t~es or g ( t ) = 2 + t for all t~s. Let us

take the sequence {a,},~l defined by

1 O.2k = 2 - -

k

1 a2k+t = ~ k = l , 2 , . . . .

Then {(Zn}~= 1 conve rges to s but the sequence {g(a , )}~ 1 has two

limit points.

CONTINUOUS SELECTIONS FROM TOPOLOGICAL TO METRIC SPACES 433

4. Applications for multifunctions with exactly n values.

LEMMA 2. Let X be a topological space. Let Y be a metric space.

Let n > 0 be an integer; Let F : X ~ Y be a l.s.c, muhifunction with

exactly n values for each z in X . Then F is Hausdorf f continuous.

Proof. Let a be an arbitrary point of X. Let e > 0 be arbitrary.

Let us denote V = B~(F(a)) and let F ( a ) = {a l , . . . , a ,~ ) . Hence

V = U B e ( { a i } ) . For i = 1 , 2 , . . . , n there exists a neighborhood Ui of i=l

a such that if t is in U~ then F ( t ) n B~({a~})~0. Denote U - - f l U ~ . i=1

Then if t is in U

(1) F(t ) n BA{ai})r for i = 1 , 2 , . . . , n

holds and since F( t ) has exactly n values we have

F(t) C B~(F(a)).

By (1) F(a) C B~(F(t)) holds. So if t is in U then H(F( t ) , F(a)) < e

Q.E.D.

LEMMA 3. Let X be a topological space. Let Y be a Hausdorf f

topological space. Let F : X ~ Y be a l.s.c, rnultifunction and let

f : X ~ Y be a continuous function. Let for each x in X the set

F ( z ) - {f(x)} be nonempry. Then the multifunction G : X ~ Y defined

by G ( x ) = F ( x ) - {f(x)} is l.s.c.

Proof. Let x be a fixed point of X. Let U be an open set in Y

and let G(x) n U~O.

First we see that there exists t in F (z ) such that t e l ( z ) , t 6 U.

Since t c f ( z ) t h e r e exist two disjoint open sets W, V in Y such that

t 6 W C U and f (x ) 6 V. Since F(x) NW#O there exists an open neighborhood 01 of z such that for each s in 01 F ( s ) n w~O. From

continuity of f it follows that there is an open neighborhood 02 of z

such that for each z in 02 f ( z ) 6 V.

434 tvAr~ K ~

Denote 0 = 01 I'~ 02. Then for each s in 0 F(8) N W#r and f ( s ) E W

therefore :r E 0 C G - ( W ) C G-(U) . Hence G - ( U ) is a neighborhood

of z. Q.E.D.

THEOREM 2. Let n > 0 be an integer. Let X be a topological

space. Let (Y,d) be a metric space. Let F : X ~ Y be a l.s.c.

multifunction with exactly n values for each z in X . Let

(v) there exists a uniformly continuous function v : Y ~ R such

that for each z in X the set v (F(z ) ) has exactly n elements (so v is

one-to-one on F(z ) ) holds.

Then there exist continuous functions fi : X --, Y for i = 1 , 2 , . . . , n

such that for each z in X F ( z ) = { f l ( z ) , . . . , f , ( z ) ) holds.

Proof (1) If n = 1 Theorem 2 is true.

(2) Let n > 0 be an integer and let us suppose that Theorem 2 is

valid for each integer k; 0 < k < n. By Lemma 2 the multifunction F

is Hausdorff continuous and the conditions of Theorem 1 are fulfilled.

Theorefore there exists a continuous selection for F . Let us denote this

selection by f l . Let us define a multifunction G : X ~ Y as follows:.

G(z) = F ( ~ r ) - { f l ( z ) } for each z in X. By Lemma 3 the

multifunction G is 1.s.c. and we can see that it has exactly n - 1 values.

Since G(x) C F(z ) for each z in X the condition (v) of Theorem 2 is

fulfilled for G. Hence, since Theorem 2 is valid for multifunctions with

n - 1 values, there exist n - 1 continuous functions f2, f 3 , . . . , f,~ from

X into Y such that for each :c in X G ( z ) = ( f 2 ( z ) , . . . , f , ( z ) } holds.

Therefore for each z in X F ( z ) = {f l ( : r ) )UG( : r ) = { f l ( z ) , . . . , f ~ ( z ) } .

Q.E.D.

REFERENCES

[1] Carbone L., Selezioni continue in spazi non lineari e punti f~ssi, Rend. Circ. Mat. Palermo 25 (1976), 101-115.

[2] Filippov A.F., Classical solutions of differential equations with multivalued right-hand side, SIAM J. Cotrol 5 (1967), 609-621.

[3] Fryszkowski A., Continuous selections for a class of non-convex multivalued maps, Studia Mathematica 76 (1983), 163-173.

CONTINUOUS S~,F_L'/'IONS FROM TOPOLOGICAL TO METRIC SPACES 435

[4] Hermes H., On continuous and measurable selections and the existence of solutions of generalized differential equations, Proc. Amer. Math. Soc. 29 (1971), 535-542.

[5] Kuratowski K., Topology, Academic press, New York, 1966.

[6] Michael E., Continuous selections I, Annals of Mathematics 63 (1956), 361-382.

[7] Parthasarathy T., Selection theorems and their applications, Lect. Notes Math. 263 (1972), 1-101.

Pcrvcnuto il 22 diecmbre 1989. in forma modifieata il 30-5-1990.

Department of Mathematical Analysis Faculty of Mathematics and Physics

Komensky University Mlynska Dolina

84215 Bratislava, Czechoslovakia