@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 351–360

Developable hyperspaces are metrizable

L’ubica Holá, Jan Pelant
∗

and László Zsilinszky

Dedicated to Professor S. Naimpally on the occasion of his 70th birthday.

Abstract. developability of hyperspace topologies (locally finite,
(bounded) Vietoris, Fell, respectively) on the nonempty closed sets is
characterized. Submetrizability and having a Gδ-diagonal in the hy-
perspace setting is also discussed.

2000 AMS Classification: 54B20.
Keywords: developable spaces, Vietoris topology, Fell topology, locally finite
topology, bounded Vietoris topology, Gδ-diagonal.

1. Introduction.

Let CL(X) (K(X)) denote the hyperspace of nonempty closed (compact)
sets of a T2 topological space (X,τ). For notions not defined in the paper see
[11], [2] and [15].

Historically there have been two hyperspace topologies of particular impor-
tance: the Vietoris topology τV (see section 1) and the Hausdorff metric topol-
ogy τH, as considered in Michael’s fundamental paper on hyperspaces [22]. It
is well-known, that τV = τH on K(X) and hence (K(X),τV ) is metrizable iff
X is. Metrizability of the larger hyperspace (CL(X),τV ) is also characterized,
it is equivalent to X being compact metrizable [22].

To investigate generalized metric properties of the Vietoris topology, one
may start by considering Bing’s factorization of metrizability into collectionwise
normality (CWN) and Mooreness ([11]). There is an abundance of results on
CWN and related properties, e.g. (combined results of Keesling and Velichko
from [19], [20], [27]): (CL(X),τV ) is CWN (paracompact, normal, resp.) iff X
is compact; however, some stronger (hereditary) properties, such as hereditary
normality, stratifiability or monotone normality, coincide with metrizability for
(CL(X),τV ) (cf. [6], [12], [13] and [7]).

∗The second author was supported by the grant GACR 201/00/1466.



352 L’ubica Holá, Jan Pelant and László Zsilinszky

As far as the other half of Bing’s Theorem is concerned, Mooreness (devel-
opability) has been considered only for K(X); indeed, Mizokami has shown
that (K(X),τV ) is Moore iff X is [23]. It is one of the purposes of this paper
to characterize developability (Mooreness) of the Vietoris topology on CL(X).
A good starting point for investigating developability is to look at 1st count-
ability of (CL(X),τV ) first, as was done by Holá and Levi [16] while extending
an older result of Choban [8]:

Theorem 1.1. Let X be a T2 space. The following are equivalent:
(i) (CL(X),τV ) is 1st countable;
(ii) X is perfectly normal, the derived set X′ is countably compact, heredi-

tarily separable and of countable character, and X \X′ is countable.

Thus, in general, 1st countability of the Vietoris topology does not guarantee
its metrizability (just consider X = ω, the discrete space of non-negative inte-
gers); other cases, e.g. if X is dense-in-itself and metrizable, are treated in [10].
In section 2 we will prove that developability and metrizability of (CL(X),τV )
always coincide. However, see Remark 3.4 following Theorem 3.3 for a relevant
comment.

In fact, after obtaining the same relationship for other well-studied hyper-
topologies, such as the locally finite, Fell and bounded Vietoris topology, re-
spectively (see section 1 for definitions), as well as for the Wijsman topology
(which is 1st countable iff it is metrizable [2]), it seems that the coincidence of
developability and metrizability in the hyperspace setting could be established
for a broad class of hypertopologies. It remains to be seen if it is the case for
hit-and-miss and hit-and-far topologies ([2]) or even for the general hyperspace
topology studied in [28], incorporating all the above topologies along with some
weak hyperspace topologies, including the Wijsman topology.

Since metrizability is to submetrizability as developability is to having a Gδ-
diagonal (see 1.6, 2.2 and 2.5 in [15]), we then turn to studying submetrizability
and having a Gδ-diagonal in CL(X). It turns out to be a perfect match for the
Fell and the bounded Vietoris topology, respectively, moreover, if X is Morita’s
M-space, also for the Vietoris topology.

Finally, in Section 4, the above generalized metric properties are discussed
on K(X) with the Vietoris and Fell topologies, respectively.

2. Preliminaries.

To describe the hypertopologies we will work with, we need to introduce
some notation: for U ⊂ X put

U+ = {A ∈ CL(X) : A ⊂ U} and U− = {A ∈ CL(X) : A∩U 6= ∅}.
Subbase elements of the Vietoris (locally finite) topology τV (τlf ) on CL(X)
are of the form U+ with U ∈ τ and

⋂
U∈U U

− with U ⊂ τ finite (locally finite).
Note, that for a metrizable X, the supremum of all Hausdorff metric (resp.
Wijsman) topologies corresponding to topologically equivalent metrics is the
locally finite (resp. Vietoris) topology ([4], [25], [5]).



Developable hyperspaces are metrizable 353

Another classical hypertopology, the Fell topology, has found numerous ap-
plications in various fields of mathematics ([21], [1]); it has as a subbase ele-
ments of the form U− and V +, where U ∈ τ and V has a compact complement
in X. If (X,d) is a metric space the bounded Vietoris topology τbVd has as a
subbase, elements of the form U− and V +, where U ∈ τ and the complement
of V is a closed bounded set in (X,d).

Proposition 2.1. Let X be a T2 space. If (CL(X),τV ) or (CL(X),τlf ) is 1st
countable, then X is collectionwise normal.

Proof. In view of Theorem 1.1 (resp. [9]), X is normal and X′ is countably
compact; thus, if D is a discrete family of closed subsets of X, only finitely
many members D0, . . . ,Dn ∈ D intersect X′. Since X is normal, there exist
pairwise disjoint open sets U0, . . . ,Un such that Di ⊂ Ui for each i ≤ n. Denote
D0 = {D0, . . . ,Dn} and observe that D =

⋃
(D\D0) is closed. Consequently,

(D\D0) ∪{U0 \D,.. . ,Un \D} is a disjoint open expansion of D. �

Proposition 2.2. Let X be a T2 space. If (CL(X),τV ) or (CL(X),τlf ) is
developable, then X is metrizable and X′ is compact.

Proof. By admissibility of the Vietoris and the locally finite topology, X is
a developable space, which is also collectionwise normal by Proposition 2.1;
hence, in view of Bing’s Theorem ([11]), X is metrizable. Since X′ is countably
compact, in our case, it is compact. �

3. Developability in CL(X).

Theorem 3.1. Let X be a T2 space. The following are equivalent:
(i) (CL(X),τlf ) is Moore;
(ii) (CL(X),τlf ) is developable;
(iii) (CL(X),τlf ) is metrizable;
(iv) X is metrizable and X′ is compact.

Proof. (ii)⇒(iv) follows from Proposition 2.2 and (iv)⇒(iii) from [4], Theorem
2.3. �

Theorem 3.2. (CL(ω),τV ) is not developable.

Proof. Suppose (CL(ω),τV ) is developable. Let D = {Un : n ∈ ω} be a
development of (CL(ω),τV ). Without loss of generality we may suppose that
Un+1 is a refinement of Un for every n. For every A ∈ [ω]ω, define

L(A) = min{n : St(A,Un) ⊂ A+},
and for every A ∈ [ω]ω and F ∈ [A]<ω put

π(A,F) = min{L(B) : F ⊂ B,B is a proper infinite subset of A}.
For every A ∈ [ω]ω, m = L(A), and every F ∈ [A]<ω choose H(A,F) ∈

[A]<ω such that F is a proper subset of H(A,F) and there is U ∈ Um with
A+ ∩

⋂
p∈H(A,F){p}

− ⊂ U.



354 L’ubica Holá, Jan Pelant and László Zsilinszky

For every A ∈ [ω]ω and every B,G ∈ [A]<ω such that H(A,B) ⊂ G we have
π(A,G) > L(A): otherwise, π(A,G) ≤ L(A) and there is an infinite proper
subset B of A with G ⊂ B such that L(B) ≤ L(A). Hence, there is U ∈UL(A)
such that {A,B} ⊂ U. As Un+1 is a refinement of Un for every n ∈ ω, we see
that for each k < L(A) there is Uk ∈ Uk with {A,B} ⊂ Uk. Hence for every
k ≤ L(A), St(B,Uk) is not subset of B+ (as A\B 6= ∅) and so L(B) > L(A),
a contradiction.

We use an inductive construction now:
• Put n0 = π(ω,∅). Take A0 ∈ [ω]ω such that L(A0) = n0 and
G0 ∈ [ω]<ω such that there is U ∈ Un0 with A

+
0 ∩p∈G0 {p}

− ⊂ U.
Put F0 = ∅.

• Let Fj+1 =
⋃
i≤j Gi and nj+1 = π(Aj,Fj+1); choose Aj+1 ∈ [ω]

ω such
that Fj+1 ⊂ Aj+1, where Aj+1 is a proper subset Aj, L(Aj+1) = nj+1
and put Gj+1 = H(Aj+1,Fj+1).

Clearly, this construction can be repeated ω-many times. Finally, put B =⋃
i∈ω Gi and take p ∈ ω such that np > L(B). Then Fp ⊂ B ⊂ Ap−1, but

L(B) ≥ π(Ap−1,Fp) = np, a contradiction. �

Theorem 3.3. Let X be a T2 space. The following are equivalent:
(i) (CL(X),τV ) is Moore;
(ii) (CL(X),τV ) is developable;
(iii) (CL(X),τV ) is metrizable;
(iv) X is compact and metrizable.

Proof. Only (ii)⇒(iv) needs justification: in view of Proposition 2.2, it suffices
to show that every sequence in X \ X′ has a cluster point in X. Otherwise,
X \X′ contains a closed copy of ω, thus, (CL(ω),τV ) sits in (CL(X),τV ) and
is hence developable, a contradiction with Theorem 3.2. �

Remark 3.4. After L’. Holá’s lecture at Caserta 2001, prof. Arhangel’skǐı
was wondering, whether Theorem 3.3 could be extended to hyperspaces which
are σ-spaces (i.e. spaces with a σ-discrete network) (see [15], also for related
notions of (strong) Σ-spaces). He was very right. A possible way could be an
easy modification of the proof of Theorem 3.2 in effect that (CL(ω),τV ) is not
a σ-space and an application of the fact that each countably compact σ-space
is compact and metrizable [15]. Fortunately, Popov [26] proved already in 1978
the following

Theorem 3.5. (CL(ω),τV ) contains the Sorgenfrey line S as a subspace.

Recall that S is not even a Σ-space. So it follows from Popov’s result that if
(CL(X),τV ) is a Σ-space then X is countably compact. Of course, Theorem
3.2 represents a special case, nevertheless we have decided to keep its proof to
make the paper more self-contained. As M-spaces are Σ-spaces, some other
information on the Vietoris hyperspaces which are Σ-spaces, may be found in



Developable hyperspaces are metrizable 355

Proposition 4.9 below. Coming back to the original Arhangel’skǐı’s question,
we could reformulate Theorem 3.3:

Theorem 3.6. Let X be a T2 space. The following are equivalent:

(o) (CL(X),τV ) has a σ-discrete network;
(i) (CL(X),τV ) is Moore;
(ii) (CL(X),τV ) is developable;
(iii) (CL(X),τV ) is metrizable;
(iv) X is compact and metrizable.

We proceed with other topologies now:

Theorem 3.7. Let (X,d) be a metric space. Then the following are equivalent:

(i) (CL(X),τbVd) is Moore;
(ii) (CL(X),τbVd) is developable;
(iii) (CL(X),τbVd) is metrizable;
(iv) (X,d) is boundedly compact (i.e. every closed bounded set in (X,d) is

compact).

Proof. Since (iii)⇔(iv) is known, only (ii)⇒(iv) needs some comments: let
B ∈ CL(X) be bounded in (X,d). Developability of (CL(X),τbVd) implies that
CL(B) equipped with the relative topology τbVd on CL(B) is also developable.
It is easy to verify that the relative topology τbVd on CL(B) coincides with
the Vietoris topology τV . Thus, (CL(B),τV ) is developable and B must be
compact by Theorem 3.3. �

Theorem 3.8. Let X be a T2 space. The following are equivalent:

(i) (CL(X),τF ) is Moore;
(ii) (CL(X),τF ) is developable;
(iii) (CL(X),τF ) is T2 and has a Gδ-diagonal;
(iv) (CL(X),τF ) is submetrizable;
(v) (CL(X),τF ) is metrizable;
(vi) X is hemicompact and metrizable.

Proof. (ii)⇒(iii) Developability of (CL(X),τF ) implies that it has a Gδ-diagonal;
moreover, even 1st countability of (CL(X),τF ) implies, that X is locally com-
pact ([16]), so, (CL(X),τF ) is T2 by a result of Fell.

(iii)⇒(v) Hausdorffness of (CL(X),τF ) implies that (CL(X),τF ) is locally
compact. Since (CL(X),τF ) has a Gδ-diagonal, points of (CL(X),τF ) are
Gδ; thus, (CL(X),τF ) is 1st countable; hence (see [17]) it is paracompact. In
summary, (CL(X),τF ) is paracompact, locally compact with a Gδ-diagonal
and is therefore metrizable.

(v)⇔(vi) is known ([3]) (to prove (vi)⇒(v) realize that every first countable
hemicompact Hausdorff space is locally compact). The remaining implications
are trivial. �



356 L’ubica Holá, Jan Pelant and László Zsilinszky

4. Submetrizability, having a Gδ-diagonal and related properties
in CL(X).

The last theorem of the previous section showed that these properties coin-
cide for (CL(X),τF ). In what follows, we show that similar relationship holds
for the (bounded) Vietoris topology as well.

First, we will see that for the (bounded) Vietoris and locally finite topology,
respectively, submetrizability and developability are distinct.

Proposition 4.1.
(i) If X is a metrizable space, then (CL(X),τlf ) is submetrizable.
(ii) If (X,d) is a separable metric space, then (CL(X),τV ) and (CL(X),τbVd)

are submetrizable.

Proof. (i) If X is a metrizable space, take any compatible metric d on X and
consider the Hausdorff metric topology τHd. It is known that τHd ⊆ τlf .

(ii) Since d is a separable metric on X, the Wijsman topology τWd is metriz-
able ([2]). It is known that τWd ⊆ τbVd ⊆ τV . �

Corollary 4.2.
(i) (CL(X),τV ) is submetrizable and not developable, if X is non-compact,

separable and metrizable.
(ii) (CL(X),τbVd) is submetrizable and not developable, if (X,d) is a sep-

arable metric space which is not boundedly compact.
(iii) (CL(X),τlf ) is submetrizable and not developable, if X is a non-compact

dense-in-itself metrizable space.

Proposition 4.3. Let X be a T2 space.
(i) If the points in (CL(X),τV ) are Gδ, then X is hereditarily separable

and every closed set in X is Gδ.
(ii) If the points in (CL(X),τF ) are Gδ, then X is hereditarily separable,

every open set in X is σ-compact and every closed set in X is a Gδ-set.
(iii) If (X,d) is a metric space, then the points in (CL(X),τbVd) are Gδ iff

(X,d) is separable.

Proof. We prove only (i): let A ⊂ X. Since A is a Gδ-set in (CL(X),τV ), there
are τV -open sets Gn such that {A} =

⋂
n∈ω Gn. Without loss of generality we

can suppose, that for every n ∈ ω, Gn = (Gn)+∩∩l≤n(Uiln )−, where Gn,Uiln , l ≤
n are open sets in X. For every n ∈ ω, l ≤ n choose ailn ∈ A∩Uiln . It is easy
to verify that {ailn : l ≤ n,n ∈ ω} = A. �

Remark 4.4. Let X be a T2 space.
(i) If (CL(X),τV ) has a Gδ-diagonal, then X is hereditarily separable.
(ii) Points in (CL(X),τV ) are Gδ iff every A ∈ CL(X) is a Gδ-set and has

a countable pseudobase (in A).

Proposition 4.5. Let (X,d) be a metric space. The following are equivalent:
(i) (CL(X),τbVd) is submetrizable;



Developable hyperspaces are metrizable 357

(ii) (CL(X),τbVd) has a Gδ-diagonal;
(iii) (X,d) is separable.

Proof. (iii)⇒(i) See Proposition 4.1(ii).
(ii)⇒(iii) Follows from Proposition 4.3(iii). �

Proposition 4.6. Let X be a w∆-space. The following are equivalent:
(i) (CL(X),τV ) is submetrizable;
(ii) X is a separable metrizable space.

Proof. (ii)⇒(i) See Proposition 4.1(ii).
(i)⇒(ii) Submetrizability of (CL(X),τV ) implies its Hausdorffness, so X is

regular by a result of Michael [22] and submetrizable, which in turn, being a
w∆-space, is an M-space ([15]). However, an M-space with a Gδ-diagonal is
metrizable ([15]). Finally, by Remark 4.4(i), X is separable. �

Proposition 4.7. Let X be an M-space. The following are equivalent:
(i) (CL(X),τV ) is submetrizable;
(ii) (CL(X),τV ) is T2 and has a Gδ-diagonal.
(iii) X is a separable metrizable space.

Proof. ((ii)⇒(i) See Proposition 4.1(ii).
(i)⇒(ii) An M-space with a Gδ-diagonal is metrizable ([15]) and by Remark

4.4(ii), X is separable. �

Proposition 4.8. Let X be a T2 space.
(i) (CL(X),τV ) is a w∆-space (strict p-space,M-space, resp.) ⇒ X is

countably compact.
(ii) X is countably compact ; (CL(X),τV ) is a w∆-space.

Proof. (i) By Proposition 4.1(ii), (CL(ω),τV ) is submetrizable, so it has a G∗δ-
diagonal ([15]). By a theorem of Hodel ([15]) this implies, that (CL(ω),τV ) is
not a w∆-space (neither is a strict p-space or an M-space, since these properties
are stronger than w∆). On the other hand, if (CL(X),τV ) is a w∆-space and X
is not countably compact, then ω sits in X as a closed subset and hence CL(ω)
embeds as a closed subset in (CL(X),τV ). Since the w∆-property is closed
hereditary, this would imply that (CL(ω),τV ) is a w∆-space, a contradiction.

(ii) By Example 2 of [24], there is a countably compact space X, such that
(F2(X),τV ) (= the space of all sets with at most 2 elements) is not a w∆-
space. Since F2(X) is a closed set in (CL(X),τV ) and the w∆-property is
closed hereditary, (CL(X),τV ) is not a w∆-space. �

A topological space X is ultracompact iff every net in X with a countable
range has a cluster point. This characterization of ultracompactness is due to
Holá and Künzi [18].

Proposition 4.9. Let X be a linearly ordered topological space. The following
are equivalent:

(o) (CL(X),τV ) is countably compact;



358 L’ubica Holá, Jan Pelant and László Zsilinszky

(i) (CL(X),τV ) is an M-space;
(ii) (CL(X),τV ) is a w∆-space;
(iii) X is countably compact.

Proof. (ii)⇒(iii) See Proposition 4.8(i).
For the rest of the proof, realize that in linearly ordered topological spaces

countable compactness and ultracompactness coincide [14], and X is ultra-
compact iff (CL(X),τV ) is [18]. Further, an ultracompact space is countably
compact, which in turn is an M-space. �

Remark 4.10. It can be inferred from the above proof, that if X is ultracom-
pact, then (CL(X),τV ) is an M-space (w∆-space). So, for every non-compact
ultracompact space X, (CL(X),τV ) is a non-compact M-space (w∆-space).

Proposition 4.8 also offers another proof of developability of (CL(X),τV ):

Proposition 4.11. Let X be a T2 space. The following are equivalent:
(i) (CL(X),τV ) is developable;
(ii) (CL(X),τV ) is a w∆-space and X has a Gδ-diagonal.

Proof. (i)⇒(ii) is clear.
(ii)⇒(i) By Proposition 4.8(i), X is countably compact. Every countably

compact space with a Gδ-diagonal is compact and metrizable [15]. Now Theo-
rem 3.3 applies. �

5. Some generalized metric properties in K(X).

Proposition 5.1. Let X be a T2 space. The following are equivalent:
(i) (K(X),τV ) is submetrizable;
(ii) X is submetrizable.

Proof. (i)⇒(ii) It is easy to verify that if T is a coarser metrizable topology on
(K(X),τV ), then the relative topology T on X is coarser than τ.

(ii)⇒(i) Let µ ⊆ τ be a metrizable topology. Then K(X,τ) (= compact sets
in τ) ⊆ K(X,µ). (K(X,µ),µV ) is a metrizable space, where µV is the Vietoris
topology generated by µ. Thus, also µV restricted to K(X,τ) is a metrizable
topology on K(X,τ). Since µ ⊆ τ, we have that µV restricted on K(X,τ) is
coarser than τV . �

The situation in (K(X),τF ) is different:

Proposition 5.2. Let X be a T2 space. The following are equivalent:
(i) (K(X),τF ) is Moore;
(ii) (K(X),τF ) is developable;
(iii) (K(X),τF ) is T2 and has a Gδ-diagonal;
(iv) (K(X),τF ) is submetrizable;
(v) (K(X),τF ) is metrizable;
(vi) X is hemicompact and metrizable.



Developable hyperspaces are metrizable 359

Proof. (iii)⇒(vi) First we show that if (K(X),τF ) is T2, then X is locally
compact. Suppose that X fails to be locally compact. Let x ∈ X be such that
for every U ∈B(x) and K ∈ K(X) there is some xU,K ∈ U\K (here B(x) stands
for a base of neighborhoods of x). Let y ∈ X be a point different from x. It is
easy to verify that the net of compact sets {{xU,K,y} : U ∈B(x),K ∈ K(X)}
converges both to {x,y} and to {y} in (K(X),τF ), a contradiction.

Since (K(X),τF ) has a Gδ-diagonal, points of (K(X),τF ) are Gδ and X has
a Gδ-diagonal. Thus, X is σ-compact, locally compact with a Gδ-diagonal, i.e.
it must be hemicompact, by [11] 3.8.C (b), and metrizable.

(vi)⇒(v) If X is hemicompact and metrizable, then (CL(X),τF ) is metriz-
able [2]; thus, (K(X),τF ) is also metrizable.

(v)⇒(iv), (iv)⇒(iii) and (i)⇒(ii) are trivial.
(ii)⇒(iii) It follows from [3], that 1st countability of (K(X),τF ) implies

hemicompactness of X and hence its local compactness (since X must be first
countable). This in turn is equivalent to Hausdorffness of (K(X),τF ).

(iii)⇒(i) By the above, (iii) is equivalent to metrizability of (K(X),τF ).
(i)⇒(ii) is trivial �

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Received January 2002

Revised October 2002

L’ubica Holá

Academy of Sciences, Institute of Mathematics, Štefánikova 49,81473 Bratislava,
Slovakia

E-mail address : hola@mat.savba.sk

Jan Pelant

Mathematical Institute, Czech Academy of Sciences, Žitná 25, 11567 Praha 1,
Czech Republic

E-mail address : pelant@cesnet.cz

László Zsilinszky

Department of Mathematics and Computer Science, University of North Car-
olina at Pembroke, Pembroke, NC 28372, USA

E-mail address : laszlo@uncp.edu