Distance functional and suprema of hyperspace topologies · topology on CL(X) such that A --. d(x, A) is continuous for each x E X is usually called the Wijsman topology in the literature

Annali di Matematica pura ed applicata (IV), Vol. CLXII (1992), pp. 367-381

Distance Functionals and Suprema of Hyperspace Topologies (*).

GERALD B E E R ( * * ) - ALOJZY LECHICKI

SANDRO LEVI(***) - SOMASHEKHAR NAIMPALLY

Abstract . - Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the wea#est topology on CL(X) such that A ~ d(x,A) is continuous for each x e X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) --~ --*inf{d(a, b): a e A, b eB} is continuous on CL(X) x CL(X).

1. - I n t r o d u c t i o n .

Let CL(X) denote the nonempty closed subsets of a metrizable topological space X. If d is an admissible metric for X, and A ~ CL(X), and x e X, then the distance from x to A is given by the familiar formula

d(x, A) =- inf d(x, a). a e A

Usually, one thinks of d as a function of the point variable, with the set A held fixed. Alternatively, we may hold x fixed to obtain a function d(x, .) on CL(X). The weakest topology on CL(X) such that A --. d(x, A) is continuous for each x E X is usually called the Wij sman topology in the literature ([16], [19]). We denote this topology by 7W(d) in the sequel. A basic question is this: what is the supremum of the Wijsman topologies, as d runs over the admissible metrics for X? Put differently, if ~ denotes the set of admissible metrics, what is the weakest topology on CL(X) such that for each x e X and each d E 0~, the functional d(x, .) is continuous on CL(X)? One main result of this

(*) Entrata in Redazione il 9 novembre 1989; versione riveduta entrata il 20 aprile 1990.

Indirizzo degli AA.: G. BEER: Department of Mathematics, California State University, Los Angeles, Los Angeles, California 90032, USA; A. LECHICKI: Hardstrasse 43, Furth D-8510, FRG; S. LEvi: Dipartimento di Matematica, Universit~ di Milano, Milano 20133, Italy; S. NAIM- PALLY: Department of Mathematics, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1.

(**) Visiting the University of Minnesota. (***) Visiting California State University, Los Angeles.

368 G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc.

paper shows that this topology is none other than the familiar Vietoris topology. On the other hand, if we restrict our attention to metrics that determine the same uniformity, then the supremum is a properly smaller topology (unless with respect to this uniformity, each continuous real valued function on X is uniformly continuous). AI- though Wijsman topologies corresponding to uniformly equivalent metrics may dif- fer [19, p. 449], this supremum is actually determined by any one of the uniformly equivalent metrics, and is functionally characterized as follows: it is the weakest topology on CL(X) such that the gap or separation functional Dd: CL(X) • CL(X) --> --~ R, defined by Dd (A, B) = inf {d(a, b): a e A, b e B}, is continuous, where d is any one of the metrics. These results show that Wijsman topologies are building blocks in the lattice of hyperspace topologies.

2 . - P r e l i m i n a r i e s .

In this section we provide basic notation, and introduce a few hyperspace topologies on CL(X). In what follows, d will be a fixed metric on X. The open (resp. closed) ball of radius ~ about x e X will be denoted by S~ [x] (resp. B~ [x]). If A c X, we denote the union of all open d-balls of radius r whose centers run over A by S~ [A]. In terms of this notation, Hausdor(f distance on CL(X) is defined by the formula

Hd (A, B) = inf {s: S~ [A] ~ B and S~ [B] ~ A}.

Such a distance defines an infinite valued metric on CL(X), which is evidently finite valued when restricted to the closed and bounded subsets of X. We denote by ZH(d) the topology induced on CL(X) by Hd.

Basic facts about Hausdorff distance can be found in [10] and in [18]. Most impor- tantly, lid (A, B) = sup I d(x, A) - d(x, B) I, so that the zH(d)-convergence of a net (A~)

x e X to A is equivalent to the uniform convergence of (d(., A~)) to d(., A). As a result, rH(d) ~ ZW(d). By equicontinuity of distance functions, the topologies agree provided (X, d} is totally bounded; the converse is also true [7,16]. The Wijsman topology zW(d) is actually metrizable if and only if X is second countable [19]; in this case, Zw(d) is also second countable. That second countability of X implies second countability and metrizability of ~W(d) follows easily from the Urysohn metrization theorem [26, p. 166]: Zw(d) is second countable and Hausdorff because it has as a subbase all sets of the form

d(x, .)- 1 (~, ~),

where x ranges over a countable dense subset of X and ~ and ,6 are rational, and it is completely regular simply because it is a weak topology. There is a great deal of literature on Wijsman convergence of sequences of sets, especially in a normed linear space. We refer the interested reader to [3, 7,23,25] and to [5].

To introduce the vietoris topology, the Fell topology, and the topology natural

G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc. 369

with respect to the gap functional Dg, we need some further notation. If A c X, the following collections of closed subsets of X are associated with A:

A - = {FeCL(X): F N A ~ O},

A + = {Fe CL(X): F c A } ,

A ++ = {Fe CL(X): Dd(F, A c) > 0} = {Fe CL(X): for some z > 0, S~[F] cA}.

Evidently, A ++ cA + and A c S~[A] ++. The Vietoris topology, also called the finite topology or the exponential topology, is generated by all sets of the form V- and V +, where V runs over the open subsets of X. We denote the Vietoris topology by Zv in the sequel (it does not vary with d). For information on this topology (undoubted- ly, the most well-studied hyperspace topology), the reader may consult the funda- mental article of MICHAEL [20], the recent monograph of KLEIN and THOMPSON [18], or [15]. The Fell topology ZF [12], also called the topology of closed convergence [18], admits a similar presentation: it is generated by all sets of the form V- where V is an open subset of X, and W + where W has compact complement. Evidently, ZF C rV; more precisely, the results of [16] show that ~. c zm(d) c Zv for each d ~ ~ . The Fell ~opology has been particularly well studied in relation to convergence and minimiza- tion problems for lower semicontinuous functions [2, 9].

A key construction in this article is the topology on CL(X) generated by all sets of the form V- and V ++, where V runs over the open subsets of X. We call this the d-proximal topology and denote it by ~o~). The motivation for this terminology and notation is as follows: sets of the form V ++ contain those closed sets that are far from V C with respect to the metric proximity associated with d (see, e.g., [21] or [26]). Since (VNW) ++= V ++ N W ++, a base for the topology ~-o~d/ consists of all sets of the form

v + + n Vl n n ... n y : ,

where V, V1, ..., V~ are open. Evidently, a local base for the topology at A ~ CL(X) consists of all sets of the form

St[A] ++ n S~[al]- n S~[az]- n ... n S~[a~]-,

where {al, a2, ..., a~} cA and ~ > 0. From this presentation, it is clear that the d-proximal topology is compatible with Fisher convergence [13, 14] of sequences of sets, as considered by BARONTI and PAPINI [3]. It is also clear that the d-proximal topology is the supremum of the lower Vietoris topology and the upper Hausdorff metric topology induced by d, as discussed in [16]. Evidently, =o~d)= ~o~/ if and only if d and p determine the same proximity (see, e.g., [21, Lemma 2.8]). In particular, if d and ~o determine the same uniformity, then Zo~)= zo~o) because a metric proximity is a special case of a proximity induced by a uniformity. In fact, the converse is true: for a metrizable space, there is a unique metric


uniformity that gives rise to a given metric proximity [21, Theorem 12.18]. In the sequel, we write d ~ ~ provided d and ~ are uniformly equivalent.

Finally we notice that z~(d) is coarser than either rv or rH(d); in the next section we show that it is finer than ~w(d). All of these facts can be shown to follow from results of [16].

3. - Suprema of Wijsman topologies.

Our f~'st result shows that the Vietoris topology is the weakest topology on CL(X) such that for each admissible metric d and each x e X, F---) d(x, F) is a continuous function on CL(X).

THEOREM 3.1. - Le t X be a metrizable space, and let ~ be the family of admissible

metrics for its topology. Then ~v = sup {~w(d): d E O)}.

PROOF. - Let z be the weak topology so described; Since zw(d) c rv for each d e 0~ ([16]), it is clear that z r To show r ; , c z, it suffices to show that V - e ~ and V + E for each open subset V or X. I t is well known that V - ~ Zw(d) for each admissible d [16, Proposition 2.1]; so V- e ~. Evidently, V + is a-open if V = X, for in this case V + = = CL(X). Otherwise, fix A e V +. We produce an admissible metric ,z and Y0 e X such

that

A ~ {B e CL(X): #(Yo, A) - 1/4 < ~(Yo, B)} c V + .

Since {B e CL(X): ~(Yo, A) - 1/4 < #(Yo, B)} e vw(~) c ~, this will show that V + contains a a-neighborhood of each of its points. Le t Yo be an arbi t rary point of V C. To obtain ~, let d be an arbi t rary admissible metric, and let ? be a Urysohn function for the sets A and V C. Define ~ as follows:

~(x, y) = min {1/2, d(x, y)} + I~(x) - ~(Y)I �9

Suppose {B e CL(X): ,z(Yo, A) - 1/4 < ,Z(yo, B)} c V + fails. Then we can find B E CL(X) such that ,O(yo, A) - 1/4 < ,~(Yo, B) and B A V C ~ ~. Choose b e B A V C ;

since ~(Yo, b) ~< 1/2, we get

1 ~< ~(Y0, A) < ~(Yo, B) + 1/4 < ,~(Yo, b) + 1/4 ~< 3/4 ,

an obvious contradiction. Thus, V + e 6, and z = ry follows. []

We next show that the d-proximal topology is the weakest topology z on CL(X) such that the gap functional ( F 1 , F2)--)Dd(F1, F 2 ) - inf{d(x, y): x e F1, y E F2} is continuous on CL(X) • CL(X).

G. BEER - A. LECHICKI - S. LEVI - S. NA1MPALLY: Distance functionals, etc. 371

THEOREM 3.2. - Let (X, d) be a metric space, and let ~ be a topology on CL(X). The following are equivalent:

(1) ~ ~ Zo~) ;

(2) D~: (CL(X), z> • (CL(X), ~> ~ R is continuous;

(3) for each A e CL(X), ~A : <CL(X), ~} --~ R defined by ~ (F) = D~ (A, F) is continuous.

PROOF. - (1) ~ (2). Let As and A2 be fLxed nonempty closed sets. We first establish upper semicontinuity of D~ at (A1, A2). To this end, fLX s > 0, and choose (as, a2) E A1 • A2 with d(ai, a2) < D~ (As, A2) + s/2. By condition (1),

=- S~/4 [al ]- • S~/4 [a2 ]- is a z-neighborhood of (A~, A2) and ff (F1, F2) ~ ~, then

Dd (F1, F~) < d(al, a2) + ~/2 < D~t (A1, A2) + ~.

Evidently, lower semicontinuity holds at (A1, A2) if Dd(A1, A z ) = 0. Otherwise, Dd (A1, A2 ) = ~ > 0. Let s e (0, ~) be arbitrary. Whenever (F1, F2) ~ S~/2 [A1 ] + + • • S~/2 [A2] + +, then for each (x, y ) e (F1, F2), we have d(x, y)1> ~ - s. As a result,

Dd (Fi, F2) I> a - s = Dd (A~, A2) - s, establishing ~-lower semicontinuity of Dd at (A1, A2 ).

(2) ~ (3). A jointly continuous bivariate function is continuous in each variable separately.

(3) ~ (1). By the remarks made in w 2, it suffices to show that S~ [x]- is in r for each x e X and each s > 0, and V + + is in z for each open subset V of X. Fix x e X and ~ > 0 ;

S~[x]- = {F ~ CL(X): Dd(F, {x}) < ~} = {F ~ CL(X): ~{~}(F) < s};

so S~ Ix]- is r-open by the z-continuity of ~{~}. On the other hand, ff V c X is open, then

V + + = {F e CL(X): ? x - v(F) > 0}

and V ++ is also r-open. Thus, z3 Zo~d). []

It is an immediate consequence of Theorem 3.2 that <CL(X), Zo~d)) is completely regular. That the hyperspace is Hausdorff follows from

COROLLARY 3.3. - Le t <X, d> be a metric space. Then ~'W(d)C Zo~d).

PROOF. - By Theorem 3.2, for each x e X, F--~ Dd ({x}, F) = d(x, F) is continuous on <CL(X), Zo~d)>. "

Note that this corollary can also be obtained from the results of[16].


EXAMPLE 3.4. - In the plane with the usual metric d, let A be the x-axis, and for each n e Z +, let A~ be the line with equation y = x / n. I t is easy to check that A = = Zw(d)-limA~. However, (A,~} fails to converge to A in the topology zo~), because i f F is the line with equation y = 1, then for each n e Z + we have Dd (A,, F) = 0, whereas

Dd(A, F) = 1.

Combining Theorem 3.1 and Corollary 3.3 we immediately obtain the following fact, which has been observed by DOLECKI and ROLEWICZ in multifunction form [11, p. 147].

COROLLARY 3.5. - Le t X be a metr~able space, and let 0) be the set of its admissible metrics. Then ~v = sup {~o~d): d e 0)}.

In conjunction with Theorem 3.2 we get a second functional characterization of the Vietoris topology.

THEOREM 3.6. - Le t X be a metrizable space. Then the Vietoris topology on CL(X) is the weakest topology on CL(X) such that the gap functional

(A1, A2) --* inf {d(al, a2): a l e A1 and a2 E A2 },

is continuous on CL(X) • CL(X)for each admissible metric d.

The last result of this section characterizes the d-proximal topology as a supremum of Wijsman topologies.

THEOREM 3.7. - Le t X be a metrizable space, and let d be an admissible metric for

X. Then sup {~w~o): ,o --- d} = ~o~d).

PROOF. - Denote sup{two): ,~ ~ d} by ~-. Suppose ,o---d. By Corollary 3.3, Zw(~) c z~), and since ,o ---- d, we have zo~o) = ':o~d). Thus, ~ C ~o~d). For the reverse inclusion, we already know that for each open V and each ?, we have V - e Zw(~) c v. Thus, it suffices to show that V + + e z for each open V. This is clearly true if V = X. Other- wise, fix A e V ++ and choose e < 1 / 2 with S2~[A]cV. We will produce ~z----d and

Y0 e X such that

(1) A c {B ~ CL(X): ~(Yo, A) - ~ < ?(Y0, B)} r St [A] + c $2~ [A] + + c V + +.

Since {B e CL(X): ;(Yo, A) - s < P(Yo, B)} e zw(;) c ~, this would prove that V + + e ~. To obtain the desired metric ~, we modify the construction presented in the proof

of Theorem 3.1. First, consider ~:X--->R defined by

,~(x)={d(x, A) ifxeS~[A],otherwise.

Since ~ is Lipschitz with constant one, it is uniformly continuous, whence the metric ,z

O. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc. 373

on Xdef ined by

,o(x, y) = rain {1/2, d(x, y)} + ~-~ ]~b(x) - ~b(y) I ,

is uniformly equivalent to d. Notice that ~(x, y) <<. 1/2 provided both x and y are in S~ [A] c, whereas if x �9 A and y �9 S~ [A] c, we have ~(x, y) I> 1.

Let Yo ~ S~ [A] C be arbi trary (S~ [A] C is nonempty because V c is). To establish the inclusions of (1) relative to p and Yo, only the inclusion

{B �9 CL(X): ?(Yo, A) - e < ,O(yo, B)} cS~[A] +

requires verification. I f this fails, then for some B �9 CL(X) we have both ~(Yo, A) - - ~ < ~(Y0, B) and B N S~ [A] c ;~ 0. Pick b ~ B N S~ [A] c; since Y0 and b are both in S~ [A] C and ~ < 1/2, we have

l~<~(yo, A)<~(yo , B ) + ~ < ; ( y o , b ) + ~ < l / 2 + e < l ,

an obvious contradiction. Thus, the inclusions of (1) are established, completing the proof. []

We have seen that the Vietoris topology is simultaneously the supremum of the Wijsman topologies and the d-proximal topologies corresponding to admissible metrics for a metrizable space X, i.e.,

"~v = s u p {vw(d): d ~ 0~} = s u p {:o~) : d �9 0~}.

What, then, is sup {ZH(d): d �9 69}? This supremum is known: it is the locally finite topology as introduced by BEER, HIMMELBERG, PRIKRY and VAN VLECK [8]. Specifi- cally, the locally finite topology on CL(X) is generated by all sets of the form V + with V open and 4 - - {.4 e CL(X): A V) V ~ 0 for each V � 9 4} where CI is a locally finite family of open subsets of X.

4. - First countability, second countability, and metrizability of the d-proximal topology.

Here, we exhibit necessary and sufficient conditions for ~o~) to be in, st countable, second countable, and metrizable. As we shall see, metrizability and second countability for this hyperspace topology are equivalent. The next lemma allows us to provide simple proofs.

LEMMA 4.1. - Let (X, d} be a metric space, and let A �9 CL(X). Then (CL(A), ":~d)) coincides with CL(A) equipped with the relative topology of (CL(X), ro~d)}.


PROOF. - Let U be A-open; there exists an X-open set V with U = V N A. We have

{F e CL(A): F N U ~ 0} = {F e CL(A): F N V ~ 0} = V- N CL(A).

Suppose Foe {F ~ CL(A): Dd (F, A - U) > 0}. I f Dd (Fo, A - U) = ~, then

FoeS~/2[Fo] ++ N CL(A) c {F eCL(A): Dd(F, A - U) > 0}.

This shows that {F e CL(A): Dd (F, A - U) > 0} contains a subspace neighborhood of each of its points. On the other hand, suppose F1 e V + + N CL(A), where V is X-open. Then for some ~, Da (F1, X - V) = ~ > 0. Let U = S~/2 [F~ ] N A; then U is A-open, and

F l e { F e C L ( A ) : D d ( F , A - U ) > O } c V + + N C L ( A ) . "

THEOREM 4.2. - Let (X, d} be a metric space. Then (CL(X), "~o~d)} is first countable if and only if X is second countable.

PROOF. - Suppose X is second countable with countable base {V~: n e Z + }. Fix F e CL(X), and let T = {n e Z + : V~ N F ~ 0}. Then all sets of the form

Svk[F] ++ N (n~T, VJ ) ,

where k E Z + and T' is a finite subset of T form a countable local base for voxd) at F. Conversely, suppose X is not second countable. Then there is an uncountable subset A of X and a positive number e such that d(al, a2) > ~ whenever al and a2 are distinct points of A. Since first countability is a hereditary property, to show that (CL(X), Zo~d)) fails to be first countable, it suffices to show, by virtue of Lemma 4.1, that (CL(A), zr is not first countable. We show that this hyperspace topology fails to have a countable base at A itself. Evidently, a local base for (CL(A), Vo~d)) at A consists of all sets of the form

~(B) =- {F e CL(A): F ~ B} = {F c A: F ~ B} ,

where B is a finite subset of A. If a countable local base exists, then there would exist finite subsets {B~: n e Z + } of A such that {D(B~): n e Z + } also forms a local base. But since A is uncountable, we can find a point ao e A - UB~, and t~({ao }) contains no t~(B~). .,

We notice that if X is not separable, {X} cannot have a countable local base for the lower Vietoris topology.

THEOREM 4.3. - Le t (X, d> be a metric space. The following are equivalent:

(1) (X, d> is totally bounded;

Cj. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: Distance functionals, etc. 375

(2) {CL(X), =o~d)} is metrizable;

(3) (CL(X), =e(d)} is second countable.

PROOF. - ( 1 ) ~ (2). If X is totally bounded, then vw(d)= 7n(d) ([7, Theorem 2.5], [16, p. 356]); so, by Corollary 3.3, we have =o~d)= :H(d). Thus, the d-proximal topology is metrizable.

(2 )~(3) . Metrizability of the hyperspace guarantees its first countability, which by Theorem 4.2, yields second countability of the underlying space {X, d}. Now ff E is a countable dense subset of X, then the set of finite subsets of E is dense in the vietoris topology for CL(X). Since ~o~) is coarser than : , , it follows that (CL(X), ro~d)} is separable. Combined with metrizability of the hyperspace, we obtain its second countability.

(3) ~ (1). Suppose d fails to be totally bounded. Then there exists an infinite subset A of X and ~ > 0 such that for all distinct a~ and a2 in A, we have d(a~, a2) > ~. By Lemma 4.1, it suffices to show that (CL(A), :o~d)} is not second countable. A base for (CL(A), ":~(d)} evidently consists of all sets of the form

I~(B, E)=- {FECL(A): B c F c E } = ~FcA: B c F c E } ,

where B is a finite subset of A and E is any subset of A. Second countability of (CL(A), void) } would ensure the existence of a base of the form {F(B,, E,~): n e Z + }. Let E be an infinite subset of A different from any E~. Consider {F ~ CL(A): Dd (F, A - E) > 0} = {F: F a E, F ~ 0}, which is nothing but E + + with respect to the space (A, d}. We claim that this set contains no I~(B~, E~ ) that contains E as an element. Clearly, the condition E ~ F(B~, E~ ) implies that E a E~. For such an E~, we see that E must be a proper subset of E~, because E ~ En. Thus, E~ ~ {F: F a E, F ~ 0}, and it follows that F(B~, E~ ) r {F: F a E, F ~ 0}. Thus, second countability of (CL(A), ~(~)} fails if (1) fails; so, (3) fails if (1) fails. []

5. - C o i n c i d e n c e o f t o p o l o g i e s .

Let (X, d} be a metric space. We have introduced six basic topologies on CL(X): the Vietoris topology =,, the Hausdorff metric topology =H(d), the Wijsman topology zwr162 the d-proximal topology ~o~), the locally finite topology rloc fro, and the Fell topology :~. The results of[16] and our work here show that vF a Zw(d)a zo~d), and that

The main purpose of this section is to completely characterize those situations in which the topologies pairwise coincide. Before we review the current state of knowl- edge, we recall a definition. A metric space (X, d} is called a UC space or an Atsuji space provided each real valued continuous function from X to R is uniformly continuous. Obviously, each compact space is UC, but there are others, e.g., any set with the zero-one metric. Among the many characterizations of UC spaces (see,

376 G. BEER - A. LECHICI(I - S. LEVI - S. NAIMPALLY: Distance functionals, etc.

e.g., [1,4,6,24, 17]), the most well-known are the following: (i) each open cover of X has a Lebesgue number; (ii) each pair of disjoint closed subsets of X lie a positive distance apart. A metrizable space admits a UC metric if and only if its set of accumula- tion points is compact [22].

I t is well-known that each of the conditions (i) rH(d) = zF, (ii) "~H(d) = ZV, (iii) ~'F :

= r v is equivalent to compactness of X. Coincidence of zw(d) and rH(d) is characterized by total boundedness of (X, d} ([7,

Theorem 2.5]). Coincidence of ZF and rW(d) occurs if and only if X has nice closed balls: each closed ball in X is either compact or is the entire space [7]. Coincidence of Zy and V~oc~n occurs if and only if X is compact [8, p. 169], whereas Zloc~m = ~H(d) if and only if (X, d} is a UC space [8, Theorem 2.2]. Since vv = r~o~ r~ implies X is compact and rF = = ~o~ f~ provided X is compact, compactness of X is equivalent to the coincidence of any

topology between ~v and zr with Z~oc~. We now look at ~o~d) and its precise relation to the others.

LEMMA 5.1. - Let (X,d} be a metric space. The following are equivalent: (a) (X, d} is totally bounded; (b) Zo~)= ZH(d); (C) ~H(d)C T V .

PROOF. - (a) ~ (b). As we mentioned in the Proof of Theorem 4.3, this follows from the result of [7] or [16] in conjunction with Corollary 3.3. We choose to provide a di- rect proof here. Given A e CL(X) and ~ > 0, we show that {F e CL(X): Hd (F, A) < ~} is in ro~d). Fix Fo with H~(Fo, A) < s. There exists ~ > 0 such that { F e CL(X): Hd(F, F0) < ~} c { F e CL(X): Hg(F, A) < ~}. By total boundedness, we can choose {xl . . . . , x~ } r F0 with S~/2 [{xl, ..., xn }] ~ Fo. Then

Fo ~ S~/~ [F0 ] § § N S~/2 Ix1 ]- (~ ... N S~/2 [xn ]- c

c {F e CL(X): Hd (F, Fo) < ~} c {F e CL(X): Hd (F, A) < ~}.

(b) ~ (c). This follows immediately from the remark at the end of w 2.

(c) ~ (a). The equivalence of (a) and (c) is a result of MICHAEL [ 2 0 , Lemma 3.2]. �9

LEMMA 5.2. - Let (X,d} be a metric space. The following are equivalent: (a) (X, d} is a UC space: (b) V~(d) = zV; (C) ZV r Vg(d).

PROOF. - ( a ) ~ (b). If (X, d} is a UC space, then for each open V, whenever F e V +, we have Dd(F, V c) > 0, i.e., F e V ++. This means that V + = V ++, so that

v~-(d) = ~v.

(b) ~ (c) This follows as in ( b ) ~ (c) above.

(c) ~ (a). The equivalence of (a) and (c) is also established in Lemma 3.2 of [20] (see also [4, Theorem 2]). m

The proof of (a) ~ (b) in Lemma 5.2 shows equally well that Zo~d) and zv agree when

G. B E E R - A . L E C H I C K I - S . L E VI - S . NAIMPALLY: D i s t a n c e f u n c t i o n a l s , etc. 3 7 7

restricted to K(X), the nonempty compact subsets of X. But rv = ZH(d) on K(X) (see, e.g., [10, p. 43] or [18, p. 41]); so, all three topologies agree here.

Our next step is to find conditions under which the d-proximal topology and the Hausdorff metric topology agree. We notice that total boundedness of (X, d) is a sufficient condition because of Proposition 2.5 of [16]. We are going to prove that it is also a necessary condition. This requires a technical lemma.

LEMMA 5.3. - Let {X, d} be a metric space. Suppose Y is an infinite uniformly isolated subset of X. Suppose for each y E Y, we have d(y, X - {y}) = d(y, X - Y). Then

TW(d) ~ Void).

PROOF. - Pick )~ > 0 such that for each y ~ Y, we have S~ [y] = {y}. Let (Yn } be a sequence of distinct terms in Y. Let A = {y~ : n ~ Z + }c and for each n e Z +, let An = = A U {y~: i I> n}. I f x ~ A, then for each n, d(x, An) = d(x, A) = 0. Otherwise, x = Yk for some k, and for each n > k we have

d(yk, A) >t d(yk, A~) = inf{d(yk, A), d(yk, {y~: i >1 n}} 1>

i> inf{d(yk, A), d(yk, Y - {Yk})} = d(yk, X - {Yk}) = d(yk, X - Y) >i d(yk, A) .

Thus, (A~} is Wijsman convergent to A, but for each index n we have Dd(A~,{yk: k E Z + } ) = 0 whereas Dd(A,{yk: k ~ Z + } ) ~ > L "

LEMMA 5.4. - Let (X,d} be a metric space that is not second countable. Then

";W(d) ~ %'(d).

PROOF. - Since X is not separable, there exists an uncountable subset E of X and > 0 such that for each x and y in E, we have d(x, y) > 3~. We consider two exhaus-

tive but not mutually exclusive cases:

(i) E contains an uncountable subset of limit points of X;

(ii) E contains an uncountable subset of isolated points of X.

In case (i), there exists n E Z + with 1 / ( n - 1) < s / 2 and a countably infinite subset ~el, e2, e~ . . . . } of E such that for each el, there exists ze~ with 1In < d(ei, ze~) <<. <~ 1/(n - 1). Let A be the following closed set:

A = S~/n[ei] �9 i = 1

Clearly, {B1/n(e~): i ~ Z +} is a discrete family of closed sets, whence for each k e Z +,

) is also a closed subset of X. We claim that A = ~w(e)-lim Ak, but A ~ ~oXd)-lim Ak. Fix

378 G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: D i s t a n c e functionals, e t c .

x �9 X. If x e A, then for each ' index k, we have d(x, A ) = d(x, A k ) = 0. Otherwise, x e S1/~[ei] for some i e Z +. For each x e S1/~[ei], we have by the triangle inequality

d(x, ze~) < 1/(n - 1) + 1In < ~.

Now, ze~ e A, and for each k ~ i, we have d(x, B1/~ [ek ]) > ~ because d(e~, ek ) > 3~. This means that for each k > i we have cl(x, A) = d(x, Ak). This proves Wijsman convergence of (Ak} to A. On the other hand, E * - {e~: i �9 Z + } is a closed set with

D d ( A , E * ) > l / n , whereas for each k e Z +, we have Dd(Ak, E * ) = 0. Thus, by the functional characterization of ro~) given in Theorem 3.2, we cannot have convergence in this topology.

We now consider the more complex case (ii). There exists ~ < z and an uncountable subset E o of E such that for each e �9 E0, we have S~ [e] = {e}. Suppose there exists a countable subset C of E0 such that for each x e Eo - C, we have d(x, X - - {x}) = d(x, C). Then with Y = E0 - C in Lemma 5.3 above, we see that ~w(d) ~ z~d). Thus it remains to show that zw(~) ~ ~o~) under condition (#):

(#) whenever C is a countable subset of Eo there exists x �9 E o - C with d(x, X - {x}) < d(x, C).

Next, we construct a sequence of distinct terms (ek) in E0 and for each k a countable subset W(k) of X such that for each k

(i) d(ek, W(k)) = d(ek, X - {ek }),

(ii) W(k) N {e~: i e Z + } = 0.

Let el ~ Eo be arbitrary, and let W(1) be an arbi trary countable (possibly of cardi- nality one) subset of X with d(e~, W(1)) = d(el, X - {e 1 }). Suppose el, e2, ..., ek and W(1), W(2), ..., W(k) have been chosen. Let ek + ~ be an arbi trary point of the uncountable set

({ klW[e~] ) E k = E o - e l , e 2 , . . . , e k } U ~ i =

such that d ( e k + l , X - {ek+l}) < inf{d(ei, ek+l): i ~</c} (such a point exists, by condition (#)). By the last inequality, we can choose a countable subset W(k + 1) of X disjoint from {e~, e2, ..., ek } with d(ek § 1, X - {ek + 1}) = d(ek + ~, W(k + 1)). By construction, the sequences (ek} and (Wk} have the asserted properties.

Now let Y = {ei: i E Z + }. For each index i, we have W(i) c X - Y; so, it is clear that d(ei, X - {ei }) = d(e~, X - Y). Applying Lemma 5.3 once again completes the proof. �9

THEOREM 5.5. - Let (X, d) be a metric space. The following are equivalent:

(1) (X,d) is totally bounded;

(2) ~w(d) = zo,(d).


PROOF. - ( 1 ) ~ (2). As mentioned earlier, total boundedness actually guarantees that 7W(d) = ZH(d).

(2) ~ (1). As outlined in w 2, :w(d) is second countable (and metrizable) provided X is second countable. Thus, if (2) holds, then ro~d) is second countable, by virtue of Lemma 5.4. By Theorem 4.3, (X,d} is totally bounded. "

We observe that the nontrivial part of Theorem 5.5 amounts to showing that total boundedness of (X, d} is necessary for the coincidence of the upper Hausdorff metric topology and the upper Wijsman topology (as defined in [16]) on CL(X).

COROLLARY 5 . 6 . - Let (X, d} be a metric space, and let (~ be the family of admissible metrics for its topology. The following are equivalent:

(1) rw(d)= rv;

(2) for each ,:~ and P2 in 0), we have r w ~ ) = "Cw(.~);

(3) :w(d) is the largest member of {Zw(e): z e 0)};

(4) X is compact.

P R O O F . - ( 2 ) ~ (3). This is trivial.

( 3 ) ~ (1). This is immediate from Theorem 3.1.

(1) ~ (4). Since a UC metric space is complete (see more generally Theorem 3 of[6]), it suffices to show that (X,d} is both UC and totally bounded. By (1) and rw(d) c Zo~) c r v , we have both rw(d) = r~(d) and ro~d) = rv. By Theorem 5.5, (X,d} is totally bounded, and by Lemma 5.2, (X,d) is UC.

(4) ~ (2). For each ,: e 0), we have zw~) = rv, provided X is compact. Thus, all the Wijsman topologies coincide. "

C O R O L L A R Y 5 . 7 . - Let (X, d) be a metric space, and let ~ = {p:~ = d }. The following are equivalent:

(1) rw(d) = ZH(d);

(2) for each p~ and ~2 in #, we have Zw(.~l) = Vw(~);

(3) Zw(d) is the largest member of {Zw~): ,z e #};

(4) (X,d} is totally bounded.

PROOF. - ( 2 ) ~ (3). This is trivial.

(3 )~(4) . By Theorem 3.7, rr sup{rw(~: ,z--d} = Zw(~). By Theorem 5.5, (X, d} is totally bounded.

(4) <~ (1). See again [7, Theorem 2.5].

(4) ~ (2). Since total boundedness is a uniform property, each ~ E ~ is totally bounded provided d is totally bounded. Also, uniformly equivalent metrics determine

380 G. BEER - A. LECHICKI - S. LEVI - S. NAIMPALLY: D i s t a n c e f u n c t i o n a l s , etc.

the same Hausdorf f metric topologies [10, p. 45]. Condition (2) now follows f rom

Theorem 2.5 of[7]. []

COROLLARY 5.8. - Le t (X, d) be a metric space. Then ro~d) = rE is and only if X is compact.

PROOF. - I f X is compact, then rE and rv agree, and Zo~d) is between the two. Con-

versely, suppose ~o~) = ZF. Then both ro~d) = zW(d) and Zw(d) = z~ hold, so that (X, d} is totally bounded, and (X ,d} has nice closed balls. Since a metric space with nice

closed balls is complete [7, p. 92], X mus t be compact. �9

The last str ing of results allows us to completely resolve the coincidence of topologies issue. We summarize our results in the following table.

Necessary and sufficient conditions for equality of hyperspace topologies.

zv = rH(d) X is compact

zv = ~,~U) (X, d} is a UC space

zv = ~W(d) X is compact

Zv = ZF X is compact

7"V = 7"loc fin X is compact

~H(d) = ~(d) (X, d) is totally bounded

7H(d) : "TW(d) (X, d) is totally bounded

rH(d) = rF X is compact

TH(d)-~ 7"1oc fin (X, d} is a UC space

Z~d) = zW(d) (X, d} is totally bounded

~'o~d) = ZF X is compact

Zo~d) = ~loe ~n X is compact

Zw(d) = ":F (X, d} has nice closed balls

~w(g) = %c ~n X is compact

r E : Tloc fm X is compact


R E F E R E N C E S

[1] M. ATSUJI, Uniform continuity of continuous functions of metric spaces, Pacific J. Math., 8 (1958), pp. 11-16.

[2] H. ATTOUCH, Variational Convergence for Functions and Operators, Pitman, Boston (1984).

[3] M. BARONTI - P. PAPINI, Convergence of sequences of sets, in Methods of Functional Analy- sis in Approximation Theory, ISNM 76, Birkhiiuser-Verlag (1986).

[4] G. BEER, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc., 95 (1985), pp. 653-658.

[5] G. BEER, On the convergence of closed sets in a metric space and distance functions, Bull. Australian Math. Soc., 31 (1985), pp. 421-432.

[6] G. BEER, More about metric spaces on which continuous functions are uniformly continuous, Bull. Australian Math. Soc., 33 (1985), pp. 397-406.

[7] G. BEER, Metric spaces with nice closed balls and distance functions for closed sets, Bull. Australian Math. Soc., 35 (1987), pp. 81-96.

[8] G. BEER - C. HIMMELBERG - K. PRIKRY - F. VAN VLECK, The locally finite topology on 2 X, Proc. Amer. Math. Soc., 101 (1987), pp. 168-171.

[9] G. BEER - P. KENDEROV, On the arg min multifunction for lower semicontinuous functions, Proc. Amer. Math. Soc., 102 (1988), pp. 107-113.

[19] C. CASTAING - M. VALADIER, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, 580, Springer-Verlag, Berlin (1977).

[11] S. DOLECKI - S. ROLEWICZ, Metric characterizations of upper semicontinuity, J. Math. Anal. Appl., 69 (1979), pp. 146-152.

[12] J. FELL, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), pp. 472-476.

[13] B. FISHER, Common fixed points of mappings and set-valued mappings, Rostock Math. Kolloq., 18 (1981), pp. 69- 77.

[14] B. FISHER, A reformulation of the Hausdorffmetric, Rostock Math. Kolloq., 24 (1983), pp. 71-76.

[15] J. FLEISCHMAN (Ed.), Set-valued mappings, selections, and topological properties of 2 X, Lecture Notes in Mathematics, 171, Springer-Verlag, Berlin (1970).

[16] S. FRANCAVIGLIA - A. LECHICKI - S. LEVI, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl., 112 (1985), pp. 347-370.

[17] H. HUEBER, On uniform continuity and compactness in metric spaces, Amer. Math. Monthly, 88 (1981), pp. 205- 205.

[18] E. KLEIN - A. THOMPSON, Theory of Correspondences, Wiley, New York (1984). [19] /L LECHICKI - S. LEVI, Wijsman convergence in the hyperspace of a metric space, Boll. Un.

Mat. Ital., 5-B (1987), pp. 435-452. [20] E. MICHAEL, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), pp.

152-182. [21] S. NAIMPALLY - B. WARRACK, Proximity Spaces, Cambridge University Press, Cambridge

(1970). [22] J. RAINWATER, Spaces whose finest uniformity is metric, Pacific J. Math., 9 (1959), pp.

567-570. [23] Y. SONNTAG, Convergence au sens de Mosco; thdorie et applications ~ l'approximation des

solutions d'indquations, Th~se d'Etat, Universit~ de Provence, Marseille (1982). [24] W. WATERHOUSE, On UC spaces, Amer. Math. Monthly, 72 (1965), pp. 634-635. [25] R. WIJSMAN, Convergence of sequences of convex sets, cones and functions, II, Trans.

Amer. Math. Soc., 123 (1966), pp. 32-45. [26] S. WILLARD, General Topology, Addison-Wesley, Reading, Mass. (1968).

Documents

Distance functional and suprema of hyperspace topologies · topology on CL(X) such that A --. d(x, A) is continuous for each x E X is usually called the Wijsman topology in the literature