Fell topology on the space of functions with closed graph - Springer Link

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo XLVIll (1999), pp. 419-430

FELL TOPOLOGY ON THE SPACE OF FUNCTIONS WITH CLOSED GRAPH I~. HOL.A - H. POPPE

Let X and Y be Tl topological spaces and G(X, Y) the space of all functions with closed graph. Conditions under which the Fell topology and the weak Fell topology coincide on G(X, Y) are given. Relations between the convergence in the Fell topology rE, Kuratowski and continuous convergence are studied too. Characterizations of a topological space X by separation axioms of (G(X, R), rF) and topological properties of (G(X, R), rE) are investigated.

1. Introduction. For topological spaces X and Y denote by C(X, Y) and G(X, Y) the spaces of all continuous functions from X to Y and of all functions from X to Y with closed graph respectively. As usual yX denotes the set of all functions from X to Y. For every f ~ yX let F f denote the graph of f i.e. F f = {(x, f ( x ) ) : x ~ X}. By K ( X ) denote the family of all compact subsets of X. In the sequel by X and Y we will always denote Tl-spaces. For a set A in X • Y denote by (A) = { f ~ yX : F f f q A = 0}. By ryk and rlyk] we mean topologies defined by the subbase elements for the open sets (K) and (K1 • K2) respectively, where K is compact in X • Y, KI is compact in X, K2 is compact in Y (see [19]).

Key words and phrases: function with a closed graph, Fell topology, compact-open topology, locally compact space.

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~:. HOL,~

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By rt we denote the lower semi-finite graph topology for yX, where rt is given by the subbase elements [G] = { f 6 yX : F f N G~0} with open sets G C X x Y (see [19]). Now put rF = vr~ V 31, r~F = rtykI V rl, where the suprema are constructed within the lattice of all topologies for yX (see [22]). rF is the Fell topology on yX and rwF is the weak Fell topology on yX. If A C X, B C Y then we will denote M ( A , B ) = { f e yX : f ( A ) C B}. The Fell topology on the hyperspace CL(X) of closed (nonempty) subsets of a topological space X was defined in [8] and then studied in many papers (see [9, 16, 17, 3, 24]). In recent years great interest arose for studying the Fell topology both in spaces of closed sets and in function spaces, stemming for instance from applications to optimization theory (see [1, 2, 4, 12, 6, 15, 23]). The graph topology ryk and its weak associated topology Y[yk] were defined in [19]. In the literature we can find also other hyperspace topologies studied on functions indentified with graphs or epigraps (see [1, 22]) as well as multifunctions (see [7]). Some results of our paper were announced in [14]. For undefined notions the reader is refered to a recent monograph [1]. In the first part of our paper we will study on the space of functions relations between the Fell topology and other known topologies as well as relations between the convergence induced from the Fell topology and known convergences (continuous, Kuratowski convergence or topological convergence) on function spaces. In the second part we consider separation axioms and other topological properties, e. g. metrizability, of the space of functions with closed graph equipped with the Fell topology.

2. The Fell topology and other topologies and convergences. In [19] was shown that if X and Y are Hausdorff locally compact spaces then ryk = r[• on C(X, Y). The following result weakens conditions on spaces X, Y and also on functions. PROPOSITION 2.1. Let X and Y be Hausdorff spaces. Then ~yk : rLykI on G(X, Y).

FELL TOPOLOGY ON THE SPACE OF FUNCTIONS WITH CLOSED GRAPH

421

Proof Let (K) be a basic r• set, and consider any f ~ (K): it suffices to find some natural number n and suitable compact sets rl

Mi

C X and Ni C Y(i = 1, 2 .... , n) such that f ~ N { M i x Ni) C (K}. i=1

Denote by A and B the projections of K on X and Y respectively and let G = F f A A x B. If G is empty we simply put n = 1, M1 = A, N1 = B and we are done; otherwise, since G and K are disjoint closed subsets of the compact space A x B, for every z = (x, y) ~ K there exists an open subset Wz of A • B which contains z and whose closure in A • B is disjoint from G. Clearly we may assume that Wz has the form Uz • Vz where Uz ~ x, Vz ~ y and Uz, Vz are open in A , B respectively. Since K is compact, there are finitely many of these open sets-say Uz~ x Vz~ (i = 1, 2 ..... n)-whose union covers K . Denote by Mi and Ni the closures o f Uzi in A and of Vzi in B, respectively: the compact set Mi x Ni is disjoint from G, and hence from l-'f, for each i = 1, 2 ..... n. Thus we have

f e r-](Mi x Ni} c N 0. For every (C, U) > (K, X) we have fc,u ~ {g E C(X, R) : [g(x) - f ( x ) [ < e Yx ~ K}); thus by Proposition 2.3 also rye-converges to f . But (x0, 1) a l i m s u p F f r , u (Let U x V be a neighborhood of (x0, 1). Let (K, G) ~ K(X) x ~(x0). For every (C, 0) > (K, G N U) we have rfc,o n ( u x v ) r [] Now we consider continuous convergence, see for instance [5], [20]. For convenience of the reader we briefly recall the definition: For topological spaces X, Y we say that a n e t (fi)icl f r o m yX converges continuously to f ~ yX, fi c f , iff for each x 6 X and for each net ( x k ) ~ x , xk ~ x implies (fi(Xk)(i.k)~l• ~ f ( x ) , where in I • coordinatewise order is used. We also can characterize continuous convergence by the use of neighborhoods (and of course by the use of filters too). Let X, Y be topological spaces, (f/)icl a net from yX, f ~ yX; then are equivalent: 1) fi

c

)f

2) For each x ~ X and for each neighborhood V of f ( x ) we find a neighborhood U of x and an index i0 6 I such that i > i0 implies y,.(U) C V. A topology r on yX (or on some subspace of yX) is called splitting iff in yX, fi c) f implies fi ~) f for each net (jr). Since the compact-open topology on yX (and on G(X, Y)) is splitting, as is wellknown, by corollary 2.3 we get that the Fell-topology is splitting on G(X, Y). But our next result shows that this topology is splitting even on yX. PROPOSITION 2.6. Let X and Y be Hausdorff spaces. Let {f~ "or

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]C} be a net in y X and f e y X . If { f , "or ~ E} continuously converges to f , then {f~ "cr ~ E} also rF-converges to f . Proof The continuous convergence of {f~ 9 o- 6 E} to f pointwise convergence and hence {f~ :6 ]C} rt-converges to f .

Further we show that the continuous convergence to f implies its rvk-convexgence. Let K C X x Y FfNK ---- 0; K is a closed set in the product X x is a Hausdorff space. For every x ~ p x ( K ) ( p x ( K ) of K to X) there are o p e n sets G x C X and H x (x, f ( x ) e G x x H x and G x x H x c ( X x Y ) \ K .

implies

of {fo 9 o" 6 E} be compact and Y since X x Y is the projection C Y such that

N o w for each x ~ p x ( K ) , by f ( x ) E H x and by the continuous convergence of {fo :~r 6 E} to f , we find an open neighborhood U x of x, U x C G x and an index Crx ~ ]C such that f o ( U x) C H x for every c r > c r x. The compactness of p x ( K ) implies that there are x l , ...xn ~ p x ( K ) such that p x ( K ) C U{U xi " i = 1,2 . . . . n}. Let cr0 E Z be such that cr >O'xi for every i 6 {1,2 .... n}. Thus for every z ~ p x ( K ) and every cr > cr0 we have (z, fo (x)) ~ K, i.e. Ffi, M K = 0 for every cr > or0), so we are done. [] Remark. In [17] was shown:

Let X and Y be topological spaces, Y Hausdorff. Let {fa :or 6 E} be a net in y X and f ~ G ( X , Y ) . If {fo : cr ~ Z} continuously converges to f , then l i m F f o = I ' f (i.e. l i m s u p F f , , C F f C l i m i n f F f ~ ) .

3. Topological properties of the Fell topology. PROPOSITION 3.1. I f X and Y are Tl-spaces then (G(X, Y), T1 too.

Z'F)

is

Proof. By the assumption the product space X x Y is TI and in X x Y the singletons are compact sets. Hence by theorem 1, remark 2 of [16] we find that the space of all nonempty closed subsets of X x Y is Tl with respect to the Fell topology and thus ( G ( X , Y), rF) is /'1 too.

If X and Y are locally compact Hausdorff spaces, then ( G ( X , Y ) , r F )

FELL TOPOLOGY ON THE SPACE OF FUNCTIONS WITH CLOSED GRAPH

425

is even Tychonoff space (see [1]). We will see that in the class of first countable spaces X, Hausdorffness of (G(X, R), rF) characterizes locally compactness of X. In what follows we will suppose that X is a Tychonoff space. We continue with the useful lemma. LEMMA 3.2. Let X be a Tychonoff space such that every singleton is a G~-set. For every finite set of distinct points Xl, x2, ..., Xn and finite set Yl, Y2 . . . . . Yn+l of reals there is a function f with closed graph such that f ( x i ) : Yi for every i ~ {1 ..... n} and f ( x ) > Yn+l otherwise.

Proof For every i 6 {1 ..... n} let fi be a continuous function from X to [0,1] such that {xi} = fi-l({0}). Put g = min{fl ..... fn}. Then g is a continuous function with {xl ..... xn} = g-l({0}). Let f be a function from X to R defined as follows: f ( x i ) = Yi, i ~ {1 . . . . . n} f ( x ) = 1/g(x) + Yn+l,

otherwise

Since g is continuous and g ( x i ) : 0 for each i w e see at once that f has a closed graph and that f satisfies the assertion of the lemma. THEOREM 3.3. Let X be a Tychonoff space such that every singleton is a Ga-set. The following are equivalent:

1) X is locally compact; 2) (G(X, ~), zr) is Tychonoff" 3) (G(X, ~), rF) is regular; 4) (G(X, ~), rF) is Hausdorff Proof (1):=~(2) X x l ~ is locally compact, thus by [1] the hyperspace C L ( X x •) of nonempty closed subsets of X x 11~ equipped with the Fell topology is Tychonoff. So also the subspace (G(X, R), rE) is Tychonoff. (2)==~(3) and (3)::~(4) are clear. (4)==~(1) Suppose there is a point x0 6 X which has no compact neighborhood.

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By L e m m a 3.2 there is a function hi with a closed graph such that

hl(xo) = 0 and h i ( x ) > 2 otherwise. Put h 2 ( x 0 ) = 1 and h z ( x ) = h i ( x ) for x~:xo. Of course also ha is a function with a closed graph. We show that hi and ha cannot be separated in ( G ( X , R ) , r F ) . So for i = 1,2 let B i be open sets in (G(X, R), rV) with hi 9 B i . Without loss of generality we can suppose that

Bi = [Ui x Hi] n [G/l] n ... N [Gimi] n (Ki) for i = 1, 2 where Ui • Hi ~ (xo, hi (x0)) and (U! • HI) N Gij = 0 for i = 1,2, j = 1,2, ...mi

(U2•

NGij=O

for i = 1 , 2 ,

j=l,2

.... mi,

Ui, Hi, Gij are open sets and Kl, K2 are compact sets.

Put K = Kl U K2.Then K is compact and the projections p x ( K ) , pR(K) are compact sets too. By the assumption there is a point xl 9 UINU2\px(K) and X2 9 UINU2\(px(K)U{xl}). NOW we choose a finite set of points (xij, h i ( x i j ) ) 9 Gij j = 1. . . . m i, i, 2. By Lemma 3.2 there is a function f with closed graph such that f ( x l ) 9 H1, f ( x 2 ) 9 /-/2, f(xij) = hi(xij) for j = 1,...,mi, i = 1,2 and f ( x ) > suppR(K) otherwise. It is easy to verify that f 9 Bl N B2. PROPOSITION 3.4. Let X be a Tychonoff space such that every sigleton is a Ga-set. If (G(X, R), rF) is first countable, then X has a coun-

table rr-base and X is hemicompact. Proof. Let f be a function identically equal to 0. Let Ul, U2 . . . . U n , . . . be a countable base of neighborhoods of f . For every n 9 Z+Un looks as follows In

Un=N[o

n xH F]N{Kn xCn).

i=1

Without loss of generality we can suppose that H/" N C. =- 0 for every i 9 {1 .... l.} and also that every O~ is not singleton; otherwise we can take K. U U{O~ 9 O~ is singleton} instead of Kn. We claim that the family {on " n 9 Z + , i 9 {1,2 .... In}} forms a rr-base which is countable of course and that the family {K. : n 9 Z +} is a countable cofinal subfamily of K ( X ) with respect to the inclusion.

FELL TOPOLOGY ON THE SPACE OF FUNCTIONS WITH CLOSED GRAPH

427

Let G be a nonempty open set in X. Then the set [G x (--E,E)] (E > 0) is a rF-neighborhood of f . Thus there must exist Un with U. C [G x ( - ~ , E ) ] . We claim there is i E {1,2 .... In} with O n C G. Suppose not. Then for every i 6 {1,2 ln} take a point xi ~ O.']\G. Let g be a function with closed graph such that g(xi) -- 0 for every, i 6 {1,2 .... , ln} and g(x) > max{supCn, E} (by L e m m a 3.2 such function g always exists). Then of course g 6 U~ but g r [G x ( - ~ , ~)]. .....

To prove hemicompactness of X let K ~ K ( X ) . Then the set (K x {1}) is a rF-neighborhood of f , so there is Uk such that Uk C / K x { l } ) . We claim that K C Kk. Suppose not. There is z ~ K \ K k . Let xi O~\{z} for every i 6 {1, 2 .... lk}. Let h be a continuous function such that h(z) = 1 and h(x) = 0 for every x 6 KkU{Xi : i ~ {1,2 .... lk}}. Then h ~ Uk\IK x {1}). PROPOSITION 3.5. Let X be a Tychonoff space such that every singleton is a Ga-set. If X is separable, then (G(X, R), rF) is separable.

Proof Let D be a countable dense set and let Q denote the rationals. By Lemma 3.2 to every A = {al ..... aj} C D consisting of distinct points, every B = {ql . . . . , , q j } C Q and every q 6 Q we can assign a function fASq with a closed graph such that fABq(ai) = qi, ai E A, qi E B and faBq(X) > q otherwise. The family {faBq : A, B finite subsets of D and Q respectively with the same cardinality, q 6 Q} is a countable dense set in (G(X, R), rF). COROLLARY 3.6 Let X be a Tychonoff space such that every singleton is a G~-set. The following are equivalent:

1. (G(X, R), rF) is metrizable; 2. (G(X, R), Z'F) is second countable and Tychonoff.

Proof (2)=*(1) is trivial (1)=*(2) By Propositions 3.4 and 3.5 (G(X, R), rF) is separable, so we have that (G(X, R), rF) is second countable since it is metrizable. COROLLARY 3.7. Let X be a Tychonoff first countable space. The following are equivalent:

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1. (G(X, 1~), rF) is metrizable; 2. (G(X, ]~), rr) is second countable and Tychonoff. 3. X is locally compact second countable. Proof (1)r is clear from the previous Corollary. ( 3 ) ~ ( 1 ) X x ]~ is locally compact second countable, thus by [1] the hyperspace C L ( X x ~) equipped with the Fell topology is metrizable. So also (G(X, JR), rF) is metrizable. (2)=:~(3) By Theorem 3.3 X is locally compact. By Theorem 4.1 in [12] the second countability of (G(X, ]~), rF) implies that X has a countable network. The local compactness of X together with the countable network guarantees the second countability of X. COROLLARY 3.8. Let X be a metrizable space: The following are equivalent:

1. X is locally compact second countable; 2. (G(X, R), rF) is second countable and Tychonoff;

3. (G(X, R), rF) is metrizable; 4. (G(X, •), rF) is first countable. Proof It is sufficient to prove (4)=~(1). The first countability of (G(X, ~), rF) gives us the hemicompactness of X and separability of X (by proposition 3.4) which imply the second countability and local compactness of X in the class of metrizable spaces.

Acknowledgement. The authors are very grateful to the referee whose valuable advices led to an improvement of our paper. Especially he pointed out that in proposition 2.1 our assumptotion that the space Y is Hausdorff and regular can be replaced by Y is Hausdorff. He also gave a simplified proof of this proposition which we used in the revised version of our paper.

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38 (1968), 89-96. [20] Poppe H., Compactness in general functions spaces, Deutscher Verlag der Wissenschaften, Berlin, 1974. [21] Poppe H., Joint continuity and compactness for a graph topology in general function spaces, Math. Nachr. 152 (1991), 173-177. [22] Poppe H., A family of graph topologies for funtions spaces and their associated weak graph topologies, Quaestiones Math. 16 (1) (1993), 111. [23] Rockafellar R., Wets R., Variation systems, an introduction in multifunctions and integrands, G. Salinetti, ed. Lecture notes in mathematies, no. 1091, Springer-Verlag, Berlin 1984. [24] Zsilinszky L., On Separation Axioms In Hyperspaces, Rend. Circ. Math. Palermo 45 (1996), 75-83. Pervenuto

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L. Hold Slovensk6 Akad~mia Vied Matematicky l)stav 81473 Bratislava Stef6nikova 49 - Slovakia H. Poppe Universitiit Rostock Fachbereich Mathematik 18055 Rostock Universitiitsplatz 1 - Germany