@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 361–376

Orderability and continuous selections for
Wijsman and Vietoris hyperspaces

Debora Di Caprio and Stephen Watson

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

Abstract. Bertacchi and Costantini obtained some conditions
equivalent to the existence of continuous selections for the Wijsman
hyperspace of ultrametric Polish spaces. We introduce a new class of
hypertopologies, the macro-topologies. Both the Wijsman topology
and the Vietoris topology belong to this class. We show that subject to
natural conditions, the base space admits a closed order such that the
minimum map is a continuous selection for every macro-topology. In
the setting of Polish spaces, these conditions are substantially weaker
than the ones given by Bertacchi and Costantini. In particular, we
conclude that Polish spaces satisfying these conditions can be endowed
with a compatible order and that the minimum function is a continu-
ous selection for the Wijsman topology, just as it is for [0, 1]. This also
solves a problem implicitely raised in Bertacchi and Costantini’s paper.

2000 AMS Classification: 54B20, 54A10, 54D15, 54E35.
Keywords: Selection, Vietoris topology, Wijsman topology, macro-topology, ∆-
topology, ordered space, compatible order, sub-compatible order, extra-dense
set, lexor, complete lexor, Polish space, star-set, n-coordinated-function, n-
coordinated-set.

1. Introduction.

We define a “macro-topology” to be an admissible hyperspace topology, finer
than the lower Vietoris topology. This new class of hypertopologies contains the
Wijsman topology, the Vietoris topology and a rich subclass of ∆-topologies.
With the help of a new object called “lexor” and using the “extra-dense” sets
(new objects as well), we construct a suitable order on a generic topological
space. We give conditions under which this order is closed, showing its “sub-
compatibility” under some further, but natural hypotheses. The properties of



362 D. Di Caprio and S. Watson

this order make it possible to prove the continuity of the minimum function
A → min(A) from a “macro-hyperspace” to the base space.

When the base space is metrizable and complete, in particular, we find a
sufficient condition for the existence of such an order, and consequently, for the
minimum map to be a continuous selection for the Wijsman hyperspace, just
as happens in the setting of compact orderable metric spaces, such as [0, 1].

The result about Wijsman hyperspaces leads to a deeper study of the con-
ditions that in [3] the authors assign on a metric space (X,d) in order to ob-
tain a continuous selection for the associated Wijsman hyperspace. As a final
result, they prove that every separable complete “ultrametric” space has a Wi-
jsman continuous selection if and only if a further condition, called “condition
(])”, holds at each point. We focus our attention on a particular subfamily of
R
P(X)\{∅}, the collection of all the real-valued functions defined on the set of

all nonempty subsets of X, whose elements are determined by a finite number
of points and real numbers. We call “n-coordinated-functions” the elements
of this family determined by n points and n real numbers. Starting on them,
we introduce the notions of “n-coordinated-set” and “star-set” (when n = 1),
showing the existence of a natural relationship between both of them and the
Wijsman basic and subbasic open subsets, respectively. Having a base of star-
sets is proved to be both a weaker condition than condition (]) given in [3],
and a stronger condition than the one developed to prove the existence of the
order.

2. Preliminaries.

Let X be a nonempty set. We denote with (X,τ) a topological space and
with (X,d) a metric space, which is understood to be endowed with the topol-
ogy induced by the metric d. Given a topology on X, let CL(X) be the
collection of all nonempty closed subsets of X and C(X,R) the set of all the
real-valued continuous functions on X.

For every E ⊆ X, E and Ec stand for the closure and the complement of E
in X, respectively. We also set:

E− = {A ∈ CL(X) : A∩E 6= ∅},

E+ = {A ∈ CL(X) : A ⊆ E} = {A ∈ CL(X) : A∩Ec = ∅}.

For every V ⊆ X, CL(X)\(V c)+ = V −.

Let (X,τ) be a topological space and ∆ be a nonempty subfamily of CL(X).
The ∆-topology τ∆ on CL(X) has as a subbase all sets of the form U−, where
U is an open set, plus all the sets of the form (Bc)+, where B ∈ ∆ (see [13]).

When ∆ = CL(X), the corresponding ∆-topology is the well-known Vietoris
topology.

Let:



Orderability and continuous selections 363

V− = {U− : U open in X},

V+ = {U+ : U open in X}.

The family V− forms a subbase for the lower Vietoris topology τ−V on CL(X);
while the family V+ determines a subbase for the upper Vietoris topology τ+V on
CL(X). The supremum of these two hypertopologies is the Vietoris topology :
τV = τ

+
V

∨
τ−V .

In general, given ∆ ⊆ CL(X), τ∆ = τ+∆
∨
τ−V , where τ

+
∆ denotes the upper

∆-topology on CL(X).

Let (X,d) be a metric space. The open ball with center x ∈ X and radius
� > 0 is given by Sd(x,�) = {y ∈ X : d(x,y) < �}. The diameter of a nonempty
subset A of X and the distance from x ∈ X to A are expressed by the familiar
formulas:

diam(A) = sup{d(a,b) : a,b ∈ A} and

d(x,A) = inf{d(x,a) : a ∈ A}.

For every x ∈ X, d(x,−) denotes the distance functional from CL(X) to R
defined by d(x,−)(A) = d(x,A) for every A ∈ CL(X). The Wijsman Topology
τWd on CL(X) is the weak topology determined by the family of distance
functionals {d(x,−) : x ∈ X}. Equivalently, the Wijsman Topology on CL(X)
can be defined by having as a subbase all the sets of the form:

A−(x,α) = {A ∈ CL(X) : d(x,A) < α} and

A+(x,α) = {A ∈ CL(X) : d(x,A) > α},

where x ∈ X and α > 0.
A net of closed subsets of X, {Aλ}λ∈Λ, τWd-converges to A ∈ CL(X) if for

every x ∈ X, limλ d(x,Aλ) = d(x,A), i.e. (CL(X),τWd) can be embedded in
C(X,R), equipped with the topology of the pointwise convergence, under the
identification map A → d(−,A) (cf. Sections 1.2 and 2.1 in [1]).

We can also present the Wijsman topology as split in two halves (Section 4.2
in [1]; see also [5], [8], [11]): τWd = τ

+
Wd

∨
τ−Wd. A subbase for τ

+
Wd

consists of
all the sets of the form A+(x,α) (x ∈ X and α > 0), whereas τ−Wd, coinciding
with the lower Vietoris topology, has as a subbase all the sets of the form U−

(U open in X).
A map f : CL(X) → X is a selection for CL(X) if f(C) ∈ C for every

C ∈ CL(X). By continuous selection we mean a selection f : CL(X) → X
also continuous with respect to the hypertopology on CL(X).

If X is a set linearly ordered by a relation <, we denote by τ< the order
topology induced by < on X, i.e. the topology having as a subbase all the rays
(←,x) and (x,→), where x ∈ X. In particular, all the intervals of the form



364 D. Di Caprio and S. Watson

(a,b), where a < b, are open with respect to τ<. Given a topology τ on X, the
order relation < on X is called compatible w.r.t. τ if τ = τ< and closed if the
set (X × X)\ ≤ is open in the product topology on X × X (≤ is the partial
order induced by <). Every compatible order is a closed order. We denote by
〈x,y〉 a point in X×X. We will also use the following terminology: <-interval
for any of the possible intervals (a,b), [a,b), (a,b] or [a,b], where a ≤ b; right
<-ray for any ray of the form (a,→) or [a,→); left <-ray for any ray of the
form (←,a) or (←,a].

3. Admissibility and macro-topologies.

Let (X,τ) be a topological space. A hyperspace topology on CL(X) is
called admissible if the relative topology induced on X by the identification
map x →{x}, coincides with the initial topology on X ([12]).

It is understood that X must satisfy some separation properties in order to
get the admissibility of the corresponding hyperspace. If (X,τ) is a T1-space,
then τ∆ is admissible (Remark 5.1 in [7]): in particular, the Vietoris topology
is admissible, whenever the base space is T1. If (X,d) is a metric space, then
the Wijsman topology on CL(X) is admissible (Lemma 2.1.4 in [1]).

Definition 3.1. Let (X,τ) be a topological space. A topology Ω on CL(X) is
called a macro-topology if it is admissible and τ−V < Ω.

If (X,τ) is a T1-space, then every ∆-topology on CL(X) is a macro-topology:
in particular, the Vietoris topology is a macro-topology. It is also clear that,
given a metric space (X,d), the relative Wijsman topology is a macro-topology.

The following will turn out to be a useful result.

Lemma 3.2. Let (X,τ) be a T1-space and H any hypertopology on CL(X)
such that the identification map x →{x} is continuous. Let B ⊆ X. If B− is
a H-closed subset of CL(X), then B is a closed subset of X.

Proof. Suppose B is not closed in X, i.e. there exists x ∈ B\B. Then there
exists a net {xα}α∈Λ of points of B converging to x. Since the identification
map is continuous, {{xα}}α∈Λ converges to {x} with respect to H. For every
α ∈ Λ, {xα}∈ B−, hence {x} must be in clH(B−) = B−, so that x ∈ B. �

Corollary 3.3. Let (X,τ) be a topological space, Ω a macro-topology on CL(X)
and B ⊆ X. If B− is a Ω-closed subset of CL(X), then B is a closed subset
of X.

Corollary 3.4. Let (X,τ) be a T1-space and B ⊆ X. If B− is a τ∆-closed
subset of CL(X), then B is a closed subset of X.

Corollary 3.5. Let (X,d) be a metric space and B ⊆ X. If B− is a τWd-closed
subset of CL(X), then B is a closed subset of X.

Remark 3.6. Let (X,τ) be a topological space and Ω a macro-topology on
CL(X). If A ∈ CL(X) and U = Ac, then the following two chains of implica-
tions are equivalent:



Orderability and continuous selections 365

• A− is Ω-closed ⇒ A is closed in X ⇒ A+ is Ω-closed;
• U+ is Ω-open ⇒ U is open in X ⇒ U− is Ω-open.

The implication U is open in X ⇒ U− is Ω-open follows from τ−V < Ω;
while A− is Ω-closed ⇒ A is closed in X has just been shown (Corollary 3.3).

4. When the minimum map is a continuous selection for
macro-hyperspaces.

Proposition 4.1. Let (X,τ) be a topological space, H be any hypertopology on
CL(X) and < be a linear order on X such that τ is generated by a family R
of right <-rays. If:

(i) for every A ∈ CL(X), min(A) exists;
(ii) for every R ∈R, R+ is a H-open subset of CL(X).

Then the mapping A → min(A) is a continuous selection from (CL(X),H)
to (X,τ).

Proof. Only the continuity of the mapping A → min(A) needs to be proved.
Let A ∈ CL(X) and R be a basic open right <-ray containing min(A). Then
R+ is H-open neighbourhood of A such that for every C ∈ R+, min(C) ∈
R. �

The dual of Proposition 4.1 is also true: we leave the proof to the reader.

Proposition 4.2. Let (X,τ) be a topological space, H be any hypertopology on
CL(X) and < be a linear order on X such that τ is generated by a family L of
left <-rays. If:

(i) for every A ∈ CL(X), min(A) exists;
(ii) for every L ∈L, L− is a H-open subset of CL(X).

Then the mapping A → min(A) is a continuous selection from (CL(X),H)
to (X,τ).

Definition 4.3. Let (X,τ) be a topological space. A linear order < on X is
called sub-compatible w.r.t. τ if τ< ≤ τ and τ has a subbase consisting of right
and left <-rays.

A topological space (X,τ) is sub-orderable (see 14.B.13 and 15.A.14 in [4])
if there exists a linear order < sub-compatible w.r.t. τ. It is well-known that
the class of sub-orderable space concides with the one of subspaces of orderable
spaces (17.A.22 in [4]). Obviously, every compatible order is sub-compatible,
just as every ordarable space is also sub-orderable.

A combination of Proposition 4.1 and Proposition 4.2 leads to the following:

Proposition 4.4. Let (X,τ) be a topological space, H be any hypertopology on
CL(X) and < be sub-compatible w.r.t. τ. If:

(i) for every A ∈ CL(X), min(A) exists;
(ii) for every subbasic open right <-ray R, R+ is a H-open subset of CL(X);

(iii) for every subbasic open left <-ray L, L− is a H-open subset of CL(X).



366 D. Di Caprio and S. Watson

Then the mapping A → min(A) is a continuous selection from (CL(X),H)
to (X,τ).

Given a sub-orderable space, conditions (ii) and (iii) of the previous propo-
sition indeed quite often can be verified: for instance if H is any ∆-topology,
where ∆ contains all the left rays which are complement of subbasic right rays
(see Lemma 3.8 in [6]); or, more concretely, if H is the Vietoris topology. After
Section 6 and the main result, it will be clear that (ii) holds true for the Wi-
jsman topology if X is a complete metric space having a countable base with
a peculiar property (see Corollary 7.2).

In case the base space is orderable, the subbasic open rays involved are of the
form (←,x) and (x,→), where x ∈ X. Moreover, if H is finer than the lower
Vietoris topology, condition (iii) is always satisfied. Therefore, the following
are immediate consequences of Proposition 4.4.

Corollary 4.5. Let (X,τ) be a topological space, Ω be a macro-topology on
CL(X) and < be a compatible linear order on X. Suppose that:

(i) for every A ∈ CL(X), min(A) exists;
(ii) for every x ∈ X, (x,→)+ is a Ω-open subset of CL(X).

Then the mapping A → min(A) is a continuous selection from (CL(X), Ω)
to (X,τ).

Corollary 4.6. Let (X,d) be a metric space whose topology admits a compatible
linear order < with respect to which each A ∈ CL(X) has a smallest element.
If for every x ∈ X, (x,→)+ is a τWd-open subset of CL(X), then the mapping
A → min(A) is a continuous selection from (CL(X),τWd) to (X,d).

Another consequence of Proposition 4.4 is a classical and well-known re-
sult about the existence of continuous selections for the Vietoris hyperspace of
orderable spaces ([9], [10]).

Corollary 4.7. Let (X,τ) be a topological space whose topology admits a com-
patible order < with respect to which each closed subset has a smallest element.
Then the mapping A → min(A) is a continuous selection from (CL(X),τV ) to
(X,τ).

The following includes an important result of Beer, Lechicki, Levi and Naim-
pally (Corollary 5.6 in [2]): we give here a more direct and easier proof.

Proposition 4.8. Let (X,d) be a metric space. The following are equivalent:
(1) X is compact;
(2) τWd = τV on CL(X);
(3) τ+Wd = τ

+
V on CL(X).

Proof. (2) ⇔ (3). It follows from the fact that τ−Wd = τ
−
V and τ

+
Wd
≤ τ+V always

happens.
(1) ⇒ (3). Let U be open in X and A ∈ U+. Uc is closed in X, and hence

compact. Also A∩Uc = ∅. Since A and Uc are closed, d(x,A) > 0, whenever
x ∈ Uc. For every x ∈ Uc, choose rx such that 0 < rx < d(x,A). The family



Orderability and continuous selections 367

{Sd(x,rx) : x ∈ Uc} is an open cover for the compact Uc. Then there exist
x1, · · · ,xn ∈ X and r1, · · · ,rn > 0 (n ∈ ω) such that Uc ⊆

⋃n
i=1 Sd(xi,ri) and

A∩Sd(xi,ri) = ∅ for every i = 1, · · · ,n. Then, A ∈
⋂n
i=1 A

+(xi,ri) ⊆ U+.
(3) ⇒ (1). If τ+Wd = τ

+
V , X is separable (see following Remark) and

(CL(X),τWd) = (CL(X),τV ) is metrizable (see Theorem 2.1.5 in [1]). By
Theorem 4.6 in [12], X is compact. �

Remark 4.9. Let (X,d) be a metric space and τδ(d) be the topology on CL(X)
generated by all the sets of the form V − and V ++ = {F ∈ CL(X) : d(F,V c) >
0}, where V runs over the open subsets of X. We can rappresent this topology
as splitted in two parts: τδ(d) = τ

+
δ(d)

∨
τ−V : it is called d-proximal topology (see

[1], [2], [8] among the others). It is known, but it is also easy to verify, that
τ+Wd ≤ τ

+
δ(d)
≤ τ+V . So, if τ

+
Wd

= τ+V , then τ
+
Wd

= τ+
δ(d)

. By Lemma 5.4 in [2] (If
(X,d) is a metric space which is not second countable, then τWd 6= τδ(d).), X
is separable.

Corollary 4.10. Let (X,d) be a compact orderable metric space. The following
are equivalent:

(a) for every x ∈ X, (x,→)+ is a τV -open subset of CL(X);
(b) for every x ∈ X, (x,→)+ is a τWd-open subset of CL(X).

Remark 4.11. It follows immediately from Corollary 4.10 that Corollary 4.6
and Corollary 4.7 are equivalent formulations of the same result for compact
orderable metric spaces.

Corollary 4.6 (or Corollary 4.7) and Proposition 4.8 yield:

Corollary 4.12. Let (X,d) be a compact orderable metric space. Then the
mapping A → min(A) is a continuous selection from (CL(X),τWd) to (X,d).

Corollary 4.13. The mapping A → min(A) is a continuous selection from
(CL([0, 1]),τWd) to ([0, 1],d), where d denotes the restriction to [0, 1] of the
Euclidean metric on R.

5. Introducing a closed linear order on X: lexors and
extra-dense sets.

We have just shown (in Section 4) how the existence of a sub-compatible
order on a topological space (X,τ) with well specified properties ((i), (ii) and
(iii) of Proposition 4.4), is a sufficient condition for the minimum map from
CL(X) to X to be a continuous selection when CL(X) is endowed with any
hypertopology.

But, when does this order exist?
Answering this question begins our main goal. This is why the present

section, which focuses on the construction of such an order, can actually be
considered as the heart of the paper.

We start with some useful definitions.



368 D. Di Caprio and S. Watson

Definition 5.1. Let X be a set. If {Un}n∈ω is a family of (arbitrary) covers
of X satisfying the property:

(I) if {An}n∈ω is a family of subsets of X such that An ∈ Un for every
n ∈ ω, then |

⋂
n∈ω An| ≤ 1,

and for every n ∈ ω, <n is a well-order (of any type) on Un, then we call
{(Un,<n)}n∈ω a lexor of X.

Definition 5.2. Let X be a set and {(Un,<n)}n∈ω be a lexor of X. For every
A ⊆ X and every n ∈ ω, let IA(n) = U if U is the <n-minimal element in
Un such that (A ∩

⋂
m<n IA(m)) ∩ U 6= ∅. If A = {x}, then we write Ix(n)

instead of IA(n): for every x ∈ X, Ix(n) = U if U is the <n-minimal element
in Un such that x ∈ U. The sequence {IA(n)}n∈ω is called the path of A, and
denoted by IA (Ix if A = {x}).

Remark 5.3. Given a set X, a lexor {(Un,<n)}n∈ω of X and A ⊆ X, the
path IA is well-defined, since each <n is a well-order.

Definition 5.4. Let X be a set and {(Un,<n)}n∈ω be a lexor of X. Given
x,y ∈ X, let ∆(x,y) = min{n : Ix(n) 6= Iy(n)}, and < be the relation on X
defined by:

(LO) x < y if Ix(∆(x,y)) <∆(x,y) Iy(∆(x,y)).
< is called the order generated by the lexor {(Un,<n)}n∈ω, and denoted by

<{(Un,<n)}n∈ω .

The terminology used in the definition above is justified by the following
proposition.

Proposition 5.5. Let X be a set and {(Un,<n)}n∈ω be a lexor of X. The
relation <{(Un,<n)}n∈ω is a linear order on X.

Proof. <{(Un,<n)}n∈ω is antisymmetric: Let x,y ∈ X, x 6= y, and suppose
that Ix(n) = Iy(n) for every n; then x,y ∈

⋂
n∈ω Ix(n); a contradiction, since

|
⋂
n∈ω Ix(n)| ≤ 1. Hence, x < y or y < x.
<{(Un,<n)}n∈ω is transitive: Let x,y,z ∈ X be such that x < y and y < z. If

∆(x,y) ≤ ∆(y,z), then ∆(x,z) = ∆(x,y) and Ix(∆(x,z)) = Ix(∆(x,y)) <∆(x,y)
Iy(∆(x,y)) ≤∆(x,y) Iz(∆(x,y)) = Iz(∆(x,z)), so that x < z. If ∆(y,z) <
∆(x,y), then ∆(x,z) = ∆(y,z) and Ix(∆(x,z)) = Ix(∆(y,z)) = Iy(∆(y,z))
<∆(y,z) Iz(∆(y,z)) = Iz(∆(x,z)), so that x < z. �

Remark 5.6. We have not yet assigned a topology on X: there is nothing
really topological in the discussion above. Moreover, the condition (I) on the
family of covers {Un}n∈ω allows us to get the antisymmetry of <. Without this
condition < would still be a linear preorder.

Definition 5.7. Let (X,τ) be a topological space. A lexor {(Un,<n)}n∈ω of
X is said to be complete if the following property holds:

(II) If A ∈ CL(X), Un ∈Un for every n ∈ ω and {Un∩A}n∈ω has the finite
intersection property, then |

⋂
n∈ω Un ∩A| 6= 0.



Orderability and continuous selections 369

Proposition 5.8. Let (X,τ) be a topological space and {(Un,<n)}n∈ω be a
complete lexor of X. Order X by <{(Un,<n)}n∈ω . Then for every A ∈ CL(X),
min(A) exists.

Proof. Fix A ∈ CL(X) and consider the path at A, IA. Then the sequence
{A ∩ IA(n)}n∈ω has the finite intersection property, and by (I) and (II),
|
⋂
n∈ω(A ∩ IA(n))| = 1. Let {xA} =

⋂
n∈ω(A ∩ IA(n)). Suppose that there

exists x ∈ A such that x < xA. Then (A∩
⋂
m<∆(x,xA)

IA(m))∩Ix(∆(x,xA)) 6=
∅, contradicting the minimality of IA(∆(x,xA)) = IxA(∆(x,xA)). Therefore,
xA = min(A). �

We also introduce the notion of “extra-dense” for subsets of an ordered
space.

Definition 5.9. Let (X,<) be an ordered space. A subset D of X is called
extra-dense if for every x,y ∈ X, x < y, D ∩ (x,y] 6= ∅.

Proposition 5.10. Let (X,τ) be a topological space and {(Un,<n)}n∈ω be a
complete lexor of X such that each initial segment of each (Un,<n) is clopen
in X. Then <{(Un,<n)}n∈ω is a closed order on X, with respect to which there
exists an extra-dense subset D with the following property:

(D) for every d ∈ D, there exists k ∈ ω and a finite sequence (U1, · · · ,Uk) ∈
U1 ×···×Uk such that (←,d) =

⋃
n≤k

⋃
V <nUn

V .

If, moreover,
⋃
n∈ω{U\

⋃
V <nU

V : U ∈Un} is a (clopen) base for τ, then τ
has as a subbase all left (<{(Un,<n)}n∈ω )-rays of the form (←,x), where x ∈ X,
and all right (<{(Un,<n)}n∈ω )-rays of the form [d,→), where d ∈ D; in particular
<{(Un,<n)}n∈ω is sub-compatible w.r.t. τ.

Proof. For x ∈ X and k ∈ ω, let L(x,k) =
⋂
n≤k(Ix(n)\

⋃
V <nIx(n)

V ). Since
the initial segments of each (Un,<n) are clopen, L(x,k) is clopen: given n ∈ ω,
an initial segment of the well-order (Un,<n) is a subset of the form

⋃
V <nU

V for
some U ∈ Un; since L(x,k) =

⋂
n≤k((

⋃
V <nS(Ix(n)) V )\(

⋃
V <nIx(n)

V )), where
S(Ix(n)) is the <n-successor of Ix(n), and by assumption,

⋃
V <nIx(n)

V and⋃
V <nS(Ix(n)) V are clopen, L(x,k) is the intersection of finitely many clopen

subsets, and hence clopen. By Proposition 5.8, L(x,k) has a minimum. More-
over, given x,y ∈ X, the following is true:

(!) if x < y, then x < min(L(y, ∆(x,y))) ≤ y.
To see this, let x < y and L = L(y, ∆(x,y)). By definition, Ix(∆(x,y)) <∆(x,y)

Iy(∆(x,y)). Obviuosly, y 6∈ Ix(∆(x,y)). Since y ∈ L, min(L) ≤ y. More-
over, for every n, min(L) must be in the minimal element of Un to intersect
L∩

⋂
m<n IL(m). Consequently, Iy(n) = Imin(L)(n) for every n ≤ ∆(x,y), so

that ∆(x,min(L)) = ∆(x,y) and x < min(L).
Let D = {min(L(x,k)) : x ∈ X,k ∈ ω}. By (!), D is an extra-dense subset

of X. To show (D), let d ∈ D. Then there exists x ∈ X and k ∈ ω such that
d = min(L(x,k)). Using (!), it is possible to check that:



370 D. Di Caprio and S. Watson

(←,d) =
⋃
n≤k

⋃
V <nIx(n)

V.

In fact, let F =
⋃
n≤k

⋃
V <nIx(n)

V . If a ∈ F, then by definition, a < x
and, by (!), a < min(L(x, ∆(a,x))) ≤ x, where ∆(a,x) ≤ k. Since L(x,k) ⊆
L(x, ∆(a,x)), min(L(x, ∆(a,x))) ≤ min(L(x,k)), so that a < min(L(x,k)).
Viceversa, if a ∈ (←,d), i.e. a < d, then Ia(∆(a,d)) <∆(a,d) Id(∆(a,d)). Notice
that a 6∈ L(x,k), otherwise min(L(x,k)) = d ≤ a. Hence, ∆(a,d) ≤ k. We
show that Ix(∆(a,d)) = Id(∆(a,d)), so that a ∈ F. It is simple to check that
for every n ≤ k, IL(x,k)(n) = Ix(n). On the other hand, since d ∈ IL(x,k)(n)
for every n, IL(x,k)(n) = Id(n) whenever n ≤ k (otherwise, there would exist
n ≤ k such that Id(n) <n IL(x,k)(n) = Ix(n) and Id(n)∩L(x,k) 6= ∅, while for
every n ≤ k and every V <n Ix(n), V ∩ L(x,k) = ∅). Hence, Ix(n) = Id(n)
for every n ≤ k. In particular, Ix(∆(a,d)) = Id(∆(a,d)).

Let x,y ∈ X such that y < x (i.e. 〈x,y〉 ∈ (X × X)\ ≤). To show the
closeness of < we need to find an open neighbourhood of 〈x,y〉 in the product
topology on X × X entirely contained in the complement of ≤. By (!), y <
d ≤ x, where d = min(L(x, ∆(x,y))) ∈ D, and L(x, ∆(x,y)) is open subset of
X containing x. By (D), (←,d) is the union of finitely many initial segments,
so it is clopen in X. Hence the set L(x, ∆(x,y)) × (←,d) is the required
neighbourhood of 〈x,y〉.

Finally, suppose that
⋃
n∈ω{U\

⋃
V <nU

V : U ∈Un} is a clopen base for the
topology on X. From the closeness of < it follows that τ< ≤ τ. In particular,
all sets of the form (a,→) and (←,a), where a ∈ X, are open in X. Moreover,
by (D), for every d ∈ D, [d,→) is clopen in X.

Now, fix x ∈ X and let U\
⋃
V <jU

V , where j ∈ ω and U ∈ Uj, be a basic
open neighbourhood of x. Note that x ∈ U\

⋃
V <jU

V if and only if U = Ix(j).
Let d = min(L(x,j)), T = (L(x,j)∪

⋃
n≤j

⋃
V <nIx(n)

V )c and t = min(T). We
claim that x ∈ [d,t) ⊆ L(x,j). Since L(x,j) ⊆ Ix(j)\

⋃
V <jIx(j)

V , this will
prove that all <-intervals [d,x), with d ∈ D and x ∈ X, form a base for τ.

Since x ∈ L(x,j), d ≤ x. On the other hand, by definition of T , ∆(x,t) ≤ j
and Ix(∆(x,t)) <∆(x,t) It(∆(x,t)), so that x < t.

Now suppose that there exists y ∈ [d,t)\L(x,j). Since d ≤ y, by (D), y ∈
[d →) = (

⋃
n≤j

⋃
V <nIx(n)

V )c. But y 6∈ L(x,j), hence y ∈ T ; a contradiction
to the fact that y < t (if y ∈ T , then t = min(T) ≤ y). �

6. The key result.

Proposition 6.1. Let (X,τ) be a T1 space and H be any hypertopology on
CL(X) such that the map x → {x} is continuous. Let {(Un,<n)}n∈ω be a
countable family of well-orders of type ≤ ω each being an open cover for X.
Suppose that:



Orderability and continuous selections 371

(i) if {An}n∈ω is a family of subsets of X with the finite intersection
propety, such that for every n ∈ ω there exists Un for which An ⊆
Un ∈Un, then |

⋂
n∈ω An| = 1;

(ii) for every n ∈ ω and for every U ∈ Un , U− is a H-closed subset of
CL(X).

Then there exists a closed linear order < on X such that:
(1) for every A ∈ CL(X), min(A) exists;
(2) X has an extra-dense subset D such that (←,d)− is a H-closed subset

of CL(X), whenever d ∈ D.
If, moreover,

⋃
n∈ω{U\

⋃
V <nU

V : U ∈ Un} is a (clopen) base for τ, then
< is sub-compatible w.r.t. τ.

Proof. By (ii) and Lemma 3.2, each cover Un consists of clopen subsets. Hence,
by (i), both (I) and (II) hold for the family {(Un,<n)}n∈ω, which is hence
a complete lexor of X. Let < be <{(Un,<n)}n∈ω . The initial segments of each
(Un,<n) are unions of finitely many clopen subsets (Un consists of clopen and
<n is of type ≤ ω). Via Proposition 5.8 and Proposition 5.10, we only need to
show (2).

Let d ∈ D. By (D), there exists k ∈ ω and a finite sequence (U1, · · · ,Uk) ∈
U1 ×···×Uk such that (←,d) =

⋃
n≤k

⋃
V <nUn

V . Consequently, (←,d)− =⋃
n≤k

⋃
V <nUn

V −, which is a H-closed subset of CL(X) by (ii). �

7. The main result.

Combining Proposition 6.1 and Proposition 4.4, we can formulate our main
result. As a consequence of it, we derive an important result related to the
problem of finding conditions, as “natural” as possible, on the base space in
order to prove the existence of continuous selections for the Wijsman hyper-
space. This will move our interest (see Section 8) to a comparison with a result
of Bertacchi and Costantini in the setting of ultrametric Polish spaces.

Theorem 7.1. Let (X,τ) be a topological space, Ω be a macro-topology on
CL(X) and {(Un,<n)}n∈ω be a countable family of well-orders of type ≤ ω
such that

⋃
n∈ω{U\

⋃
V <nU

V : U ∈Un} is a base for τ and each Un is an open
cover for X. Suppose that:

(i) if {An}n∈ω is a family of subsets of X with the finite intersection
propety, such that for every n ∈ ω there exists Un for which An ⊆
Un ∈Un, then |

⋂
n∈ω An| = 1;

(ii) for every n ∈ ω and for every U ∈ Un , U− is a Ω-closed subset of
CL(X).

Then there exists a sub-compatible linear order < on X such that the map-
ping A → min(A) is a continuous selection from (CL(X), Ω) to (X,τ).

Corollary 7.2. Let (X,τ) be a complete metrizable space and Ω be a macro-
topology on CL(X). If X has a countable base B such that B− is Ω-clopen,
whenever B ∈ B, then there exists a sub-compatible linear order < on X such



372 D. Di Caprio and S. Watson

that the mapping A → min(A) is a continuous selection from (CL(X), Ω) to
(X,τ).

Proof. For i ∈ ω, let Bi = {B ∈ B : diam(B) < 1i } (the diameters are taken
with respect to a compatible metric on X). Order each Bi by a well-order <i of
type ≤ ω. Each Bi is a countable open cover for X, satisfying (ii) of Theorem
7.1, while

⋃
i∈ω{B\

⋃
V <iB

V : B ∈ Bi} is a base for X. Since X is complete,
the family {Bi}i∈ω satisfies (i) of Theorem 7.1, as well. �

Corollary 7.3. Let (X,d) be a complete metric space having a countable base
B such that B− is τWd-clopen, whenever B ∈ B. Then there exists a sub-
compatible linear order < on X such that the mapping A → min(A) is a
continuous selection from (CL(X),τWd) to (X,d).

8. Coordinated-functions, coordinated-sets and star-sets.

Dealing with the problem of finding conditions equivalent to the existence
of continuous selections for the Wijsman topology, Bertacchi and Costantini
introduce the following notion (see Definition 2 in [3]), which turns out to play
a fundamental role in such a research:

Definition 8.1. Let (X,d) be a metric space and x ∈ X. We say that the
condition (]) holds at x if for every � > 0 there exist δ,θ ∈ R, with 0 < δ <
θ ≤ �, such that Sd(x,δ) = Sd(x,θ).

A Polish space is a separable completely metrizable space. The main result
of Bertacchi and Costantini (Theorem 3 in [3]) can be written as follows:

Proposition 8.2. Let (X,d) be a Polish ultrametric space. The following are
equivalent:

(a) condition (]) holds at each x ∈ X;
(b) there exists a continuous selection from (CL(X),τWd) to (X,d).

In this section, we give some conditions weaker than condition (]): under
such conditions each Polish space is proved to have a countable base satisfying
the requirement of Corollary 7.2. We can then conclude, without caring about
ultrametric properties, that a continuous selection not only always exists for
Wijsman hyperspaces, but that there is a very natural map which is a selection,
namely, the minimum map.

We introduce the following notion:

Definition 8.3. Let (X,d) be a metric space, x1, · · · ,xn ∈ X and r1, · · · ,rn
> 0 (n ∈ ω). The n-coordinated-function determined by x1, · · · ,xn, r1, · · · ,rn
is the function frixi : P(X)\{∅} → R defined by f

ri
xi

(A) = max(ri −d(xi,A)),
for every nonempty A ⊆ X. We write coordinated-function instead of 1-
coordinated-function.

Definition 8.4. Let (X,d) be a metric space and V be a proper subset of
X. V is a n-coordinated-set if there exist x1, · · · ,xn ∈ X and r1, · · · ,rn > 0
(n ∈ ω) such that



Orderability and continuous selections 373

(i) frixi (a) ≥ 0 for every a ∈ V ;
(ii) frixi (V

c) < 0.

A coordinated-set is a 1-coordinated-set.

For the next definition we need the notion of “excess”. Given a metric space
(X,d) and A,B ⊆ X, the excess of A over B with respect to d is defined by
the formula

ed(A,B) = sup{d(a,B) : a ∈ A}.

Excess may assume value +∞ and is not symmetric (see Section 1.5 in [1]
for more details and examples). In particular, ed(A,x) will denote the excess of
the set A over the singleton {x}. Notice that ed(x,A) = d(x,A), while ed(A,x)
is quite different.

Definition 8.5. Let (X,d) be a metric space and V be a proper subset of X.
V is a star-set if there exists xV ∈ X, such that:

(∗) ed(V,xV ) < d(xV ,V c).
We say that V is a star-set around x.

Lemma 8.6. Let (X,d) be a metric space, x ∈ X and V be a proper subset of
X. V is a coordinated-set if and only if it is a star-set.

Proof. If V is a coordinated-set, there exist x ∈ X and r > 0 such that
frx(a) ≥ 0 for every a ∈ V and frx(V c) < 0, i.e. r ≥ d(x,a) if a ∈ V and
r < d(x,V c). Hence d(a,x) ≤ r < d(x,V c) whenever a ∈ V . Therefore,
ed(V,x) = sup{d(a,x) : a ∈ V} < d(x,V c).

Suppose now that V is a star-set and that (∗) holds for some x. Let r =
ed(V,x). Then, d(x,a) ≤ ed(V,x) = r < d(x,V c) for every a ∈ V . The
coordinated-function frx satisfies (i) and (ii) of the definition of coordinated-
set. �

Lemma 8.7. Let (X,d) be a metric space and x ∈ X. The following are
equivalent:

(a) condition (]) holds at x;
(b) for every � > 0 there exists a positive δ < � such that Sd(x,δ) is a

star-set around x;
(c) inf{r > 0 : Sd(x,r) is a star-set around x} = 0.

Proof. (a) ⇒ (b). Given � > 0, there exist δ, θ ∈ R such that 0 < δ < θ ≤ � and
Sd(x,δ) = Sd(x,θ). It is easy to check that ed(Sd(x,δ),x) < d(x,Sd(x,δ)

c), so
that (∗) holds at x.

(b) ⇒ (c) and (c) ⇒ (a) are easy to check. �

Corollary 8.8. Let (X,d) be a metric space. If condition (]) holds at each
x ∈ X, then X has a base B of star-sets of the form Sd(x,δ).



374 D. Di Caprio and S. Watson

Proof. Suppose condition (]) holds at each point. By Lemma 8.7, for every
x ∈ X and � > 0, there exists δ� < � such that Sd(x,δ�) is a star-set. Since
{Sd(x,�) : x ∈ X,� > 0} is a base for the topology on X, so is the family B =
{Sd(x,δ�) : x ∈ X,� > 0}. �

Proposition 8.9. Let (X,d) be a separable metric space. If condition (]) holds
at each x ∈ X, then X has a countable base B consisting of star-sets.
Proof. By Corollary 8.8, X has a base A consisting of star-sets. Since X
is separable, it also has a countable base. Hence, there exists a countable
subfamily B ⊆ A, which is still a base for X (in general, if X is a regular
second countable space and B is a base for X, then there exists a countable
subcollection B′ ⊆B which is again a base for X). �

Lemma 8.10. Let (X,d) be a metric space, V an open subset of X and n ∈ ω.
The following are equivalent:

(1) V is an n-coordinated-set;
(2) (V c)+ =

⋂n
i=1 A

+(xi,ri), where xi ∈ X and ri > 0 for every i.
Proof. Let x1, · · · ,xn ∈ X and r1, · · · ,rn > 0. The following are equivalent:

• frixi (a) ≥ 0 for every a ∈ V and f
ri
xi

(V c) < 0;
• for every a ∈ V there exists i such that d(xi,a) ≤ ri, and d(xi,V c) > ri

for every i;
• for every A ∈ CL(X), A∩V = ∅ if and only if d(xi,A) > ri for all i;
• (V c)+ =

⋂n
i=1 A

+(xi,ri).
�

Let (X,τ) be a topological space and A ⊆ X. A is a basic closed subset of
X if Ac is a basic open subset.

Corollary 8.11. Let (X,d) be a metric space and V an open subset of X.
Then:

(i) V is a star-set if and only if (V c)+ is a τ+Wd-subbasic open subset of
CL(X);

(ii) V is a n-coordinated-set, for some n ∈ ω, if and only if (V c)+ is a
τ+Wd-basic open subset of CL(X);

(iii) V is a n-coordinated-set, for some n ∈ ω, if and only if V − is τ+Wd-basic
closed subset of CL(X).

In particular,
(iv) if V is a n-coordinated-set, for some n ∈ ω, then (V c)+ and V − are

τWd-clopen subsets of CL(X).

Proof. Recall that V − = CL(X)\(V c)+ (see Preliminaries) and V − is open in
the Wijsman topology, if V is open (Remark 3.6). �

Remark 8.12. From Corollary 8.11(iv), it follows immediately that if (X,d)
has a base B consisting of n-coordinated-sets (n ∈ ω), then (Bc)+ and B− are
τWd-clopen subsets of CL(X), for every B ∈B.



Orderability and continuous selections 375

We close with the preannounced result.

Proposition 8.13. Let (X,d) be a Polish space having a countable base of
star-sets. Then there exists a sub-compatible order on X such that the mapping
A → min(A) is a continuous selection from (CL(X),τWd) to (X,d).

Proof. Let B be the given base of star-sets. By Corollary 8.11(iv), B− is τWd-
clopen, whenever B ∈B. Apply Corollary 7.3. �

The fact that condition (]) is sufficient for the Wijsman hyperspace to admit
a continuous selection, stated by Bertacchi and Costantini in the setting of
ultrametric spaces (see (a) ⇒ (b) of Proposition 8.2), follows now as an easy
consequence and without requiring d to be an ultrametric. This also solves the
problem which is implicitely raised in the last two lines of [3].

Corollary 8.14. Let (X,d) be a Polish space such that condition (]) holds at
each point of X. Then there exists a continuous selection from (CL(X),τWd)
to (X,d).

Proof. By Proposition 8.9, X has a countable base of star-sets. Apply Propo-
sition 8.13. �

Notice that the converse of Proposition 8.13 holds if (X,d) is an ultrametric
space: use (b) ⇒ (a) of Proposition 8.2 and Proposition 8.9.

This yields to the following result, which completes the main one of [3] and
shows that the converse of Proposition 8.9 is also valid provided that d is an
ultrametric.

Proposition 8.15. Let (X,d) be a Polish ultrametric space. The following are
equivalent:

(a) X has a countable base of star-sets;
(b) there exists a sub-compatible order on X such that the minimum map

A → min(A) is a continuous selection from (CL(X),τWd) to (X,d);
(c) there exists a continuous selection from (CL(X),τWd) to (X,d);
(d) condition (]) holds at each x ∈ X.

Proof. (a) ⇒ (b) follows from Propositon 8.13; (b) ⇒ (c) is trivial; (c) ⇒ (d) is
(b) ⇒ (a) of Proposition 8.2; (d) ⇒ (a) follows from Proposition 8.9. �

Acknowledgements. The authors wish to thank C. Costantini for re-
marking that Corollary 8.14 actually solves the problem left open in [3] and
suggesting Proposition 8.15, as well as the comments before it.



376 D. Di Caprio and S. Watson

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Received February 2002

Revised October 2002

Debora Di Caprio

Department of Mathematics and Statistics, York University, 4700 Keele Street,
North York, Ontario, Canada, M3J 1P3

E-mail address : dicaper@mathstat.yorku.ca

Stephen Watson

Department of Mathematics and Statistics, York University, 4700 Keele Street,
North York, Ontario, Canada, M3J 1P3

E-mail address : watson@mathstat.yorku.ca