数理解析研究所講究録
第 1987 巻 2016 年 55-66

55

Topological structures of hyperspaces of compact
sets

$/$

finiteKatsuhisa Koshino*
Faculty of Engneering,
Kanagawa University

Introduction

1

In this article, we discuss the topological structures of hyperspaces of compact or finite sets inmetrizable spaces. In infinite-dimensional topology, it is one of the most important problems
to determine the homeomorphism types of hyperspaces. The main purpose of this article
is to provide necessary and suﬃcient conditions on spaces whose hyperspaces of compact or
finite sets are homeomorphic to pre-Hilbert spaces. This article is a r\’esum\’e of the papers[7, 8].
Throughout this article, any space is assumed metrizable and any map is assumed continuous. We let denote an infinite cardinal and often consider it as a space endowed withthe discrete topology. Let Comp(X) be the hyperspace of non-empty compact sets in a space$X$ with the Vietoris topology, and let) $\subseteq Comp(X)$ be the hyperspace of non-empty$X$
we denote the Hilbert space of density
By
finite sets in with the relative topology.
we denote the linear subspace spanned by the canonical orthonormal basis, and by. The hyperspace Comp(X) is a classical object in infinite-dimensional topology andofhas been studied by many mathematicians. By the eﬀorts of D.W. Curtis, R.M. Schori and
J.E. West, the following celebrated theorem was established:$\kappa$$Fin\langle X$$\ell_{2}(\kappa)$$l_{2}^{f}(\kappa)$$\kappa$$l_{2}(\kappa)$

Theorem 1.1. For a space $X$ , the hyperspace Comp(X) is homeomorphic to the Hilbert cube
if and only if $X$ is non-degenerate, connected, locally connected and compact.
D.W, Curtis [2, 3] obtained some characterizations of the hyperspace Comp(X) for a
non-compact space $X$ . Especially, he characterized a space $X$ whose hyperspace Comp(X)is homeomorphic to the separable Hilbert space as follows:
if
Theorem 1.2. Let $X$ be a space. The hyperspace Comp(X) is homeomorphic toand only if $X$ is separable, connected, locally connected, topologically complete and nowhere
locally compact.
$l_{2}(\omega)$$*$

This work was supported by JSPS KAKENHI Grant Number $15K17530.$

56
Concerning the hyperspace Fin(X), D.W. Curtis and N.T. Nhu [5] gave a necessary andsuﬃcient condition on $X$ for Fin(X) to be homeomorphic to
as follows:$\ell_{2}^{f}(\omega)$

Theorem 1.3. For a space $X$ , the hyperspace Fin(X) is homeomorphic to
if and only
$ifX$ is non-degenerate, connected, locally path-connected, strongly countable-dimensional and$\ell_{2}^{f}(w)$$\sigma$-compact.

Recall that a space $X$ is strongly countable-dimensional if it is a countable union of finite
dimensional closed subsets. We say that a space $X$ islocally)compact if it is written asa countable union of (locally )compact subsets. By an $X$ -manifold we understand a spacein which each point has a neighborhood homeomorphic to an open subset of a space $X.$
Recently, K. Mine, K. Sakai and M. Yaguchi [18, 9] proved the following:$\sigma-($

Theorem 1.4. For a connected.
to$\ell_{2}(\kappa)$-manifold$X$, the hyperspace Comp(X) is homeomorphic$\ell_{2}(\kappa)$

Theorem 1.5. If a space $X$ is a connected
homeomorphic to.

$\ell_{2}^{f}(\kappa)$-manifold, then the hyperspace Fin(X) is$\ell_{2}^{f}(\kappa)$

In this article, we generalize their results as follows:

Theorem A. Let $X$ be a space. The hyperspace Comp(X) is homeomorphic to
if andonly ifconnected, locally connected, topologically complete, nowhere locally compact, andevery non-empty open subset of $X$ is of density$P_{2}(\kappa)$$X\dot{u}$$\kappa.$

Theorem B. For a space $X$ , the hyperspace Fin(X) is homeomorphic to
if and only
$X$ is connected,
locallypath-connected,
strongly countable-dimensional -locally compactifand each non-empty open set in $X$ is of density$P_{2}^{f}(\kappa)$$\sigma$$\kappa.$In the last section, we will deal with the topological type of pair of hyperspaces. Given
spaces $X$ and , by $(X, Y)$ we understand is a subspace of $X$ . A pair $(X, Y)$ of spaces ishomeomorphic to
if there is a homeomorphism : $Xarrow X’$ such that $f(Y)=Y’.$subset of a space $X$ is called locally non-separating in $X$ if for any non-empty connected
open set in $X,$
is non-empty and connected. As a corollary of Theorems A andthe following can be established:
$Y$

$Y$

$(X\prime, Y’)$

$f$

$A$

$A$

$U$

$U\backslash A$$B,$

Corollary. Let $X$ be a connected, locally path-connected, strongly countable-dimensionaland -locally compact space whose any non-empty open subset is of density . Supposethat $X$ admits a locally connected and nowhere locally compact completion X. Then thepair (Comp(X), Fin(X)) of hyperspaces homeomorphic to
if and only if the$\sigma$$\kappa$$(P_{2}(\kappa),P_{2}^{f}(\kappa))$$\dot{u}$remainder2$X\backslash X$

is locally non-separating in X.

Characterizations ofand$\ell_{2}(\kappa)$$\ell_{2}^{f}(\kappa)$

In this section, we shall introduce characterizations of pre-Hilbert spaces
andusedin the proofs of Theorems A and B. For maps : $Xarrow Y$ and : $Xarrow Y$ , and for an open$\ell_{2}(\kappa)$

$f$

$g$

$\ell_{2}^{f}(\kappa)$

57
containing $f(x)$of $Y,$ is -close to if for each point $x\in X$ , there exists
and $g(x)$ . A space $X$ has the countable locally finite approximation property if the following
condition holds:

cover$\bullet$$f$$\mathcal{U}$$U\in \mathcal{U}$$\mathcal{U}$$g$, there is a sequence
For each open coverevery
is -close to the identity map on $X$ and the collection
finite in $X.$$\{f_{n} :$\mathcal{U}otX$$f_{n}$

Xarrow X\}_{n<\omega}$$\mathcal{U}$It is sa\‘id that a space $X$ has the -discrete -cells property, where
provided that the following condition is satisfied:$n$$l\sigma$

is locallyis a non-negative integer,

For every open cover of $X$ and every rrxap : $[O, 1]^{n}\cross\kappaarrow X$ , there exists a map
issuch that is -close to and the family
discrete in $X.$$f$$\mathcal{U}$$\bullet$$n$of maps such that$\{f_{n}(X)\}_{n<\omega}$’$g$$\{g([O, 1]^{n}\cross\{\gamma\})\}_{\gamma<\kappa}$$f$$\mathcal{U}$$1]^{n}\cross\kappaarrow X$

$g$

.
and
This property plays an important role in characterizing non-separable[1,(cf.
H. Toru\’{n}czyk [14, 15] gave the following elegant characterization to Hilbert spaces
Theorem 3.1]):$P_{2}^{f}(\kappa)$$l_{2}(\kappa)$

Theorem 2.1. A space $X$ is homeomorphic to$\ell_{2}(\kappa)$

if and only if the following conditions

are satisfied:$(1\rangle X$

is a topologically complete $AR$ of density ;$\kappa$(2) $X$ has the countable locally finite approximation property;(3) $X$ has the -discrele -cells property for every $n<\omega.$$n$$\kappa$

A closed subset
is satisfied:
$0$

$A$

of a space $X$ is said to be$a$(strong) $Z$ -set in $X$ if the following condition

For each open cover of $X$ , there is a map : $Xarrow X$ such that) misses $A.$the identity map on $X$ and the (closure of)image$f$$\mathcal{U}$$f$exists -close to$\mathcal{U}$$f\langle X$

We define a space $X$ to be strongly universal for a class$\bullet$$C$

if the following condition holds:

Let $A\in C,$ be a closed subset of , and : $Aarrow X$ be a map such that the restriction
is an embedding and $f(B)$ is a -set in $X$ . For any open cover of $X$ , there is anembedding : $Aarrow X$ such that is -close to $f,$ $g(A)$ is a -set in $X$ , and $g|_{B}=f|_{B}.$
$A$

$B$

$f$

$Z$

$f|_{B}$$\mathcal{U}$$Z$$\mathcal{U}$

$g$

$g$

3. Mogilski [10] characterized the separable pre-Hilbert space
Theorem 2.2.
hold:$\mathcal{A}$space $X$ is homeomorphic to(1) $X$ is a strongly countable-dimensional$\sigmaarrow$$\ell_{2}^{f}(\omega)$$\ell_{2}^{f}(\omega)$

as follows:

if and only if the following conditions

compact A ;$R$(2) $X$ is strongly universal for the class of $finitearrow$ dimensional compact metrizable spacess58(3) every finite-dimensional compact subset of $X$ is a strong$Z$-set in $X.$

The above theorem was generalized by K. Sakai and M. Yaguchi [12]:Theorem 2.3. A space $X$ is homeomorphic to$\ell_{2}^{f}(\kappa)$

are satisfied

if and only if the following conditions(1) $X$ is a strongly countable-dimensional, -locally compact $AR$ of density , and is written
as a countable union of strong $Z$ -sets,$\sigma$$\kappa$(2) $X$ is strongly universal for the dass of strongly countable-dimensional locally compact
spaces of density$\leq\kappa.$

Using the -discrete -celk property, the author [6] modified their result as follows:$n$

Theorem 2.4. A space $X$ is homeomorphic to$\ell_{2}^{f}(\kappa)$

hold:(1) $X$ is a strongly countable-dimensional(2)$X\dot{u}$$\sigma$

if and only if the following conditions-locally compact $AR$ of density$\kappa_{f}.$strongly universal for the class of finite-dimensional compact metrizable spaces,(3) every finite-dimensional compact subset of $X$ is a strong $Z$ -set in $X$(4) $X$ has the -discrete -cells property for every $n<\omega.$$\kappa$

3$n$Basic facts on Comp(X) and Fin(X)

In this section, we shall list some basic properties of the hyperspaces Comp(X) and Fin(X).
The following propositions are well-known, refer to [11, Theorem 5.12.5 (2)] and [9, Proposition 5.3]:

Proposition 3.1. A space $X$ is topologically complete if and only if so is the hyperspaceComp(X).

Proposition 3.2. For any space $X,$ $X$ is strongly countable-dimensional and -locally compact if and only if so is Fin (X) .
$\sigma$

For the sake of convenience, by Hyp(X) we mean the hyperspace Comp(X) or Fin(X).
Concerning the density of Hyp(X), the following holds, see [18, Corollary 4.2] and [9, Proposition 5.3]:

Proposition 3.3. The hyperspace Hyp(X) has the same density as a space $X.$

We can easily observe the following proposition:
Proposition 3.4. For a space $X$ , if all non-empty open sets in Hyp(X) are of density
then so are all non-empty open subsets of $X.$$\geq\kappa,$

59
On the ANR-property of Comp(X), the following holds [17, 13] (cf. [3, Theorem 1.6]):
locally connected (connected
Proposition 3.5. Let $X$ be topologically complete. Then
$ANR(AR)$.
is anand locally connected) if and only
$X\dot{u}$$\grave{\iota}fComp(X)$Combining Lemmas 2.3 and 3.6 of [5] with the proof of Theorem 2.4 in [5] (cf. [18,Proposition 3.1]), the following can be proven:
Proposition 3.6. A space $X$ is locally path-connected (connected and locally path-connected)
if and only if Fin(X) is an $ANR(AR)$ .

The countable locally finite approximation property
and the -discrete -cells property in Comp(X) andFin(X)

4$n$$\kappa$

This section is devoted to detecting the countable locally finite approximation property andthe -discrete -cells property in the hyperspaces Comp(X) and Fin(X). For a non-negative
be the -dimensional unit ball.
be the -dimensional unit sphere and let
irzteger , let
The following lemma established in [5, Lemma 3.3] is very useful for constructing maps frompolyhedra1 to hyperspaces.
$n$$\kappa$$S^{n}$$n$$B^{n}$$n$$n$

Lemma 4.1. For each $n\geq 1$ , there is a “retraction $r:B^{n+1}arrow Fin(S^{n})$ such that $r(x)=$for all $x\in S^{n},$$\{x\}$

Given a simplicial complex $K$ , we denote the polyhedron of $K$ by $|K|$ and the $n-$-skeleton
as a simplicial complex consisting offor each $n<\omega$ . We often regardof by
its faces. Now we prove the following lemma, that is a generalization of the technique usedin the proof of [2, Theorem ].
$\sigma\in K$$K^{(n)}$

$K$

$E$

Lemma 4.2. Let $X$ be a locally path-connected space. Suppose that every non-empty open
. Then the hyperspace Hyp(X) satisfies
set in $X$ contains a discrete subset of cardinality
the following.
$\geq\kappa$$\cdot$$\bullet$. For each, and $K$ be a discrete union of
be a simplicial complex,Letopen cover of Hyp(X) and each map : $|K|=\oplus_{\gamma<\kappa}|K_{\gamma}|arrow Hyp(X)$ , there is a
is locally finite-close map $h:|K|arrow Hyp(X)$ to such that the family
in $Hyp(X)$ .
$K_{\gamma}s$$\gamma<\kappa$$K_{\gamma}$$\mathcal{V}$$\mathcal{V}$$g$$\{h(|K_{\gamma}|)\}_{\gamma<\kappa}$$g$

Sketch of proof.
$\bullet$$\bullet$Replace $K$ with a suﬀciently small subdivision.

For each $n<\omega$ , choose a locally finite open cover

1Polyhedra are not needed to be metrizable.

$\mathcal{V}_{n}$of $X$ of mesh $<1/n.$

60$\bullet$

Each non-empty $V\in\nu_{n}$ contains a discrete subset
such that, whereif$Z^{\gamma}(n)\cap Z^{\tau}(n)=\emptyset$(1) Construction of$\bullet$

For every$\gamma\neq\tau$$Z(V)=\{z_{V}^{\gamma}\}_{\gamma<\kappa}$

of cardinality$Z^{\gamma}(n)=\{z_{V}^{\gamma}|\emptyset\neq V\in \mathcal{V}_{n}\},$$\kappa$$\gamma<\kappa.$$h|_{K^{(0)}}$$v\in K_{\gamma}^{(0)}$, there are$n_{w}\geq 2$and$z^{\gamma}(v)\in Z^{\gamma}(n_{v})$such thatis suﬃciently close to $g(v)$ ;-for any $u,$ $u’\in K^{(0)},$ $n_{u}=n_{u’}$ if $g(u)$ is suﬃciently close to $g(u’)$ .
$-z^{\gamma}(v)$$\bullet$

Let $h(v)=g(v)\cup\{z^{\gamma}(v)\}.$(2) Construction of$\bullet$

For each$m\leq 2\}$$\bullet$$h|_{|K(1)}|$andis the barycenter of$\sigma\in K_{\gamma}^{(1)}\backslash K_{\gamma}^{(0)}$, where$\hat{\sigma}$$\sigma^{(0)}=\{v_{1},v_{2}\}$, define$h(\hat{\sigma})=g(\hat{\sigma})\cup\{z^{\gamma}(v_{m})|1\leq$$\sigma.$

Taking an arc: $[0, 1]arrow X$ from some point oftosmall diameter, we define a path
: $[0, 1]arrow Hyp(X)$ from$\alpha_{m}$$g(\hat{\sigma})$$z^{\gamma}(v_{m})$$\phi$

of suﬃciently
toby$g(\hat{\sigma})$$h(\hat{\sigma})$$\phi(t)=g(\hat{\sigma})\cup\{\alpha_{m}(t)|1\leq m\leq 2\}.$$\bullet$

For each $m\in\{1$ , 2 , definc
betweenand , as follows:$h:\langle v_{m},$ $\hat{\sigma}\ranglearrow Hyp(X)$$\}$, where$\langle v_{m},$$\hat{\sigma}\rangle$

is the segment$\hat{\sigma}$$v_{m}$$h((1-t)v_{m}+t\hat{\sigma})=\{\begin{array}{ll}g((1-2t)v_{m}+2t\hat{\sigma})\cup\{z^{\gamma}(v_{m})\} if 0\leq t\leq 1/2,\phi(2t-1)\cup\{z^{\gamma}(v_{m})\} if 1/2\leq t\leq 1.\end{array}$(3) Construction of$\bullet$$h$Assume that$\bullet$extends overfor some $n<\omega$ such that for every
for some $A\in Fin(|\sigma^{(1)}|)$ .
For each, there exists a map, whereboundary of , such that $r(y)=\{y\}$ for all
by Lemma 4.1.
For each, let $h(y)= \bigcup_{y\in r(y)}h(y’)$ .
$K^{(n)}\backslash K^{(0)},$$h$$\bullet$$y\in\sigma\in$$h(y)= \bigcup_{a\in A}h(a)$$\sigma\in K^{(n+1\rangle}\backslash K^{(n)}$$\sigma$$\bullet$$|K^{(n)}|$$r:\sigmaarrow Fin(\partial\sigma)$$\partial\sigma$

is the$y\in\partial\sigma$$y\in\sigma$

Replace $h(y)$ with $g(y)\cup h(y)$ , that is the desired map.

$\square$We will verify the countable locally finite approximation property:
Proposition 4.3. Let $X$ be a locally path-connected and nowhere locally compact space. Then
Hyp(X) has the countable locally finite approximation property.

Sketch of proof.
$\bullet$$\square$Approximate Hyp(X)$\cross\omega$by a polyhedron $|K|=\oplus_{n<\omega}|K_{n}|$ , and apply Lemma 4.2.

61
We shall introduce useful lemmas for recognizing the -discrete -celIs property in spaces.
$n$$\kappa$

Lemma 4.4 (Lemma 3.1 of [1]). Let $n<\omega.$ A space $X$ has the -discrete -cells property
if and only if the foltowing condition holds:$n$$\kappa$$\bullet$

For each open cover$g:[0, 1]^{n}\cross\kappaarrow X$ toand each map
such thatof$\mathcal{U}$

$f$

$X$

$f$

: $[0, 1]^{n}x\kappaarrow X$ , there is a -close map
is locally finite in $X.$$\mathcal{U}$$\{g([O, 1]^{n}\cross\{\gamma\})\}_{\gamma<\kappa}$

Lemma 4.5 (Lemma 3.2 of [1]). For $n<\omega$ , a space $X$ with the countable locally finite
approximation property has the -discrete -cells property if and only if $X$ has the -discrete-cells property for anyof uncountable cofinality.
$\lambda$$\kappa$$n$$\lambda\leq\kappa$$n$

The next lemma follows from the proof of [1, Lemma 6.2]:

Lemma4.6. Suppose that has uncountable cofinality. Any space $X$ of density
a closed discrete subset of cardinality$\kappa$$\geq\kappa$

contains$\geq\kappa.$

Now we prove the following:
Proposition 4.7. Let $X$ be a locally path-connected and nowhere locally compact space. If, then the hyperspace Hyp(X) has the
any non-empty open subset of $X$ is of density-discrete -cells property for every $n<a/.$$\geq\kappa$$n$$\kappa$

Sketch of proof.
$\bullet$$\bullet$

The hyperspace Hyp(X) has the countable locally finite approximation property by
Proposition 4.3.

We may assume that is of uncountable cofinality due to Lemma 4.5, and hence any
by Lemma 4.6.
non-empty open set in $X$ contains a discrete subset of cardinality$\kappa$$\geq\kappa$$\bullet$

According to Lemma 4.4, it suﬃces to prove that any map $g:[0, 1]^{n}\cross\kappaarrow Hyp(X)$ canisbe ppproximated by a map $h,$ $[0, 1]^{n}\cross\kappa\prec Hyp(X)$ such that
locally finite in Hyp(X).
$\{h([O, 1]^{n}\cross\{\gamma\})\}_{\gamma<\kappa}$$\bullet$niangulate$[O, 1]^{n}\cross\kappa$

into a polyhedron$|K|=\oplus_{\gamma<\kappa}|K_{\gamma}|$, and use Lemma 4.2.

$\square$5

The strong universality and the compact-strong $Z-$
set property of Fin(X)This section is devoted to verifying the strong universality of Fin(X) for the class of finitedimensional compact metrizable spaces and the strong -set property of compact sets inFin(X). D.W. Curtis and N.T. Nhu showed the following:$Z$

62
Lemma 5.1 (Lemma 4.2 of [5]). Let $X$ be a locally path-connected space with finitely many
components, at least one of which is non-degenerate, and let $A\subseteq Fin(X)$ be separable.
Suppose that each member ofintersects all components of $X$ andhas the following
expansion property:$\mathcal{A}$For each $F\in Fin(X)$ , if$\bullet$Then$\mathcal{A}$$F$$\mathcal{A}$contains some element of , then$\mathcal{A}$$F\in \mathcal{A}.$is strongly universal for the class of finite-dimensional compact metrizable spaces.

Using this lemma, we show the strong universality of a non-separable hyperspace Fin(X).

Proposition 5.2. $IfX$ is a non-degenerate, connected, locally path-connected space, then thehyperspace Fin(X) is strongly universal for the class offinite-dimensional compact metrizable
spaces.

Sketch of proof.
For any finite-dimensional compact metrizable space and any map
regardas a closed subset of $[0, 1]^{n}$ for some $n<\omega$ , and extend$A$$\bullet$$A$$[0, 1]^{n}arrow Fin(X)$

$f$
$f$

: $Aarrow Fin(X)$ ,to a map :$\tilde{f}$.

The set $A=$ { $F\in Fin(\cup\tilde{f}([0,1]^{n}))|F$ contains some element of} is strongly
universal for the class of finite-dimensional compact metrizable spaces by Lemma 5.1.
$\tilde{f}([0,1]^{n})$$\bullet$$\square$

The following lemma is useful for detecting strong -sets in ANRs.
$Z$

Lemma 5.3 (Lemma 7.2 of [4]). Suppose that is a topologically complete and closed subset
of an $ANR$ X. If is written as a countable union of strong -sets in $X$ , then it is a strong$\mathcal{A}$

$A$

$Z$

$Z$

-set.

For each $k<\omega$ , the set Fi$n^{}$$(X)=$Fin(X).

{ $A\in Fin(X)|$$A$

is of cardinality$\leq k$} is closed in

Proposition 5.4. Let $X$ be a non-degenerate, connected and locally path-connected space.
Then for every $k<\omega$ , Fi (X) is a strong -set in Fin(X).
$Z$$n^{}$

Sketch of proof.
$\bullet$$\bullet$Approximate Fin(X) by a polyhedron $|K|$ whose simplexes are suﬃciently small.
It remains to show that for each map : $|K|arrow Fin(X\rangle$ , there exists a map : $|K|arrow$
Fin(X) suﬃciently close to such that the closure of $h(|K|)$ dose not meet Fi (X) .
$g$

$g$

(1) Construction of$\bullet$-$n^{}$$h|_{K(0)}$Find $k+1$ points $z(v, 0)$ ,$h$$\cdots,$$z(v, k)\in X$ for each $v\in K^{(0\rangle}$ so thateach $z(v,j)$ ) is suﬃciently close to $g(v)$ ;63$-z(v,j)s$ keep a suﬃcient distance from each other.
$\bullet$

Let $h(v)=g(v)\cup\{z(v,j)|0\leq j\leq k\}.$(2) Construction of$\bullet$$h$Follow the same argument as in Lemma 4.2.

$\square$Combining Lemma 5.3 with Proposition 5.4, we can obtain the following proposition:
Proposition 5.5. If $X$ is non-degenerate, connected and locally path-connected space, thenevery compact set in Fin(X) is a strong -set.
$Z$

Proofs of Theorems A and6$B$

In this section, applying the Characterizing Theorems 2.1 and 2.4, we proves Theorems
and B.

Proof of Theorem$\bullet$

$A$

$A.$

The case that $X$ is separable follows from Theorem 1.2, so we may only consider rc
uncountable.

(1) The く ‘only if” part$\bullet$

According to opositions 3 , 3.3 and 3.5, Comp(X) is a topologically complete
AR of density
The countable locally finite approximation property of Comp(X) follows from
Proposition 4.3.
By Proposition $4.7_{\}}Comp(X)$ has the -discrete -cells property for every $n<\omega.$$P_{1}\cdot$$.1$$\kappa.$$\bullet$$\bullet$$\bullet$$\kappa$$n$

Use the Toru\’{n}czyk Characterization Theorem 2.1.

(2) The “if” part$0$$\bullet$Due to Propositions 3.1 and 3.5, $X$ is connected, locally co1mected and topologically complete.
It follows from Proposition 3,4 that all non-empty open subsets of $X$ are of density, and hence $X$ is nowhere locally compact.
$\kappa$$\square$

Proof of Theorem$\bullet$$B.$

The separable case follows from Theorem 1.3, so we can assume$\kappa$

to be uncountable.

64(1) The “only if”’ part$\bullet$By virtue of Propositions 3.2 and 3.6, Fin(X) is a strongly countable-dimensionaland -locally compact AR of density
Due to Proposition 5.2, Fin(X) is strongly universal for the class of finite-dimensional
compact metrizable spaces.
$\kappa.$$\sigma$$\bullet$$\bullet$

According to Proposition 5.5, any finite-dimensional compact subset of Fin(X) isa strong -set in Fin(X).
$Z$$\bullet$$\bullet$

It follows from Proposition 4.7 that Fin(X) has the -discrete -cells property ofFin(X) for each $n<\omega.$
Use Theorem 2.4.
$n$$\kappa$(2) The (if” part$\bullet$$\bullet$By Propositions 3.2 and 3.6, $X$ is connected, locally path-connected, strongly
countable-dimensional and a-locally compact.
Any non-empty open set in $X$ is of density by Proposition 3.4.
$\kappa$$\square$7

Pair of hyperspaces

In this final section, we will discuss the topological structure of pair of hyperspaces. To show
Corollary, we will use the following $characterizatio\iota 1$ of the pair:$(P_{2}(\kappa),\ell_{2}^{f}(\kappa))[16, 6]$Theorem 7.1. A pair $(X, Y)$ of spaces is homeomorphic to
if and only if $X$ is,homeomorphic tois homeomorphic toand is homotopy dense in $X.$$(\ell_{2}(\kappa),l_{2}^{f}(\kappa))$$l_{2}(\kappa)$

$Y$

$Y$

$\ell_{2}^{f}(\kappa)$

of a space $X$ is defined to be homotopy dense in $X$ if there is a homotopy$h:X\cross[O, 1]arrow X$ such that $h(x, 0)=x$ for all $x\in X$ and $h(X\cross(O, 1])$. The homotopy
density between ANRs is characterized as follows [11, Corollary 7.4.6]:

A subset$A$$\subseteq A$

Lemma 7.2. Suppose that $X$ and are ANRs anddense in X. Then$X$
dense in if and only if the following condition is satisfied:$Y$$\bullet$$Y\dot{u}$$Y$

is homotopy

For each point $x\in X$ and each neighborhood of in $X$ , there exists a neighborhoodof such that any map : $S^{n}arrow V\cap Y$ can be extended to a map$U$$V\subset U$$U\cap Y$

for all$x$$\tilde{f}:B^{n+1}arrow$

$f$

$x$

$n<\omega.$

Using this lemma, we shall prove the following proposition:
Proposition 7.3. Let $X$ be a locally path-connected space that admits a locally connected
locally
completion X. Then Fin(X) is homotopy dense in Comp
if and only if
non-separating in X.
$(\overline{X})$$\overline{X}\backslash X\dot{u}$65
Sketch of proof.
(1) The (only if”’ part$\bullet$Follow the same argument as the proof of implication
rem 3.2].

$(ii)\Rightarrow(\ddot{u}i)$in [4, Theo-(2) The (if part$\bullet$$\bullet$and Fin(X) are ANRs by Propositions 3.5 and 3.6.
take
For each point $A\in Comp(\overline{X})$ and each neighborhood $u$ of in$a_{m}\in A$ and connected open neighborhoodsofa finite number of points$1\leq i\leq n$ , so that;
$B\in
Comp(\overline{X})$ ,for allif
and-for any
The hyperspaces Comp$(\overline{X})$$A$$Co\alpha tp(\overline{X})$)$a_{1},$$U_{i}$$\cdots,$$a_{i},$$-A \subset\bigcup_{i=1}^{n}U_{i}$$B\in \mathcal{U}$$B c\bigcup_{i=1}^{n}U_{i}$$B\cap U_{1}\prime\neq\emptyset$$i\in\{1, \cdots , n\}.$$\bullet$The set{is a neighborhood of$\mathcal{V}=$$\mathcal{B}\in Comp(\overline{X})|B\subset$$\mathcal{A}$Uk$U_{i}$

and$B\cap U_{i}\neq\emptyset$

for every $1\leq i\leq n$ }as in Lemma 7.2.

[3

Combining the above proposition with Theorems $A,$$B$

and 7.1, we can prove Corollary.

References[1] T. Banakh and I. Zarichnyy, Topological groups and convex sets homeomorphic to nonseparable Hilbert spaces, Cent. Eur. J. Math. 6:1 (2008), 77-86.

[2] D.W. Curtis, Hyperspaces homeomorphic to Hilbert space, Proc. Amer. Math. Soc. 75,(1979), 126-130.
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Faculty of Engineering
Kanagawa University
Yokohama, 221-8686, JapanE–mail: $pt120401we@kanagawa_{r}\cdot u.ac.jp$