HYPERSPACES OF SEPARABLE BANACH SPACES WITH THE WIJSMAN TOPOLOGY

WIESŁAW KUBIŚ, KATSURO SAKAI, AND MASATO YAGUCHI

Abstract. Let X be a separable metric space. By Cld_{W} (X), we denote the hyperspace of non-empty closed subsets of X with the Wijsman topology. Let Fin_{W} (X) and Bdd_{W} (X) be the subspaces of Cld_{W} (X) consisting of all non-empty finite sets and of all non-empty bounded closed sets, respectively. It is proved that if X is an infinite-dimensional separable Banach space then Cld_{W} (X) is homeomorphic to (≈) the Hilbert space ℓ_{2} and Fin_{W} (X) ≈ Bdd_{W} (X) ≈ ℓ_{2} × ℓ_{2}^{f}, where

ℓ_{2}^{f} = {(x_{i})_{i∈ℕ} ∈ ℓ_{2} | x_{i} = 0 except for finitely many i ∈ ℕ}.

Moreover, we show that if the complement of any finite union of open balls in X has only finitely many path-components, all of which are closed in X, then Fin_{W} (X) and Cld_{W} (X) are ANR’s. We also give a sufficient condition under which Fin_{W} (X) is homotopy dense in Cld_{W} (X).

1. Introduction

Let Cld(X) be the set of all non-empty closed sets in a topological space X. By Cld_{V} (X) we denote the space Cld(X) with the Vietoris topology, which is the most typical topology. The Curtis-Schori-West Hyperspace Theorem is a celebrated result in Infinite-Dimensional Topology which states that Cld_{V} (X) is homeomorphic to (≈) the Hilbert cube Q = [−1, 1]^{ω} if and only if X is a non-degenerate, connected and locally connected compact metrizable space ([8] and [21]; cf. [14, Theorem 8.4.5]). For a noncompact metric space X, since Cld_{V} (X) is non-metrizable, we have to consider topologies different from the Vietoris topology. For a study of hyperspace topologies, we refer to the book [4] (cf. [13]).

In the paper [17], it is shown that the space Cld_{F} (X) with the Fell topology is homeomorphic to Q \ {0} if and only if X is a locally compact, locally connected separable metrizable space with no compact components. In [2], it is shown that if X is an infinite-dimensional Banach space with weight w(X), then the space Cld_{AW} (X) with the Attouch-Wets topology is homeomorphic to a Hilbert space with weight 2^{w(X)}. It should be remarked that Cld_{AW} (X) = Cld_{F} (X) for a finite-dimensional normed linear space X (cf. [4, p.144]). In [11], the first author proved that if X is an infinite-dimensional separable Banach space, then the space Cld_{W} (X) with the Wijsman topology is an AR. In this paper, we prove the following:

1991 Mathematics Subject Classification. 54B20, 54C55, 57N20.

Key words and phrases. Hyperspace, Wijsman topology, Banach space, Hilbert space, ℓ_{2} × ℓ_{2}^{f}, AR, homotopy dense, Lawson semilattice.

The first author was supported by KBN Grant No. 5P03A04420.

Theorem I. If X is an infinite-dimensional separable Banach space, then Cld_{W} (X) is homeomorphic to the separable Hilbert space ℓ_{2}.

Let Fin(X) ⊂ Cld(X) be the set of all non-empty finite sets. By Fin_{V} (X), Fin_{W} (X), etc., we respectively denote the subspaces of Cld_{V} (X), Cld_{W} (X), etc. In [7], it is proved that Fin_{V} (X) ≈ ℓ_{2}^{f} if and only if X is a nondegenerate, strongly countable-dimensional,^{1} connected, locally path-con-nected, σ-compact metrizable space, where

ℓ_{2}^{f} = {(x_{i})_{i∈ℕ} ∈ ℓ_{2} | x_{i} = 0 except for finitely many i ∈ ℕ}.

In [17], it is shown that Fin_{F} (X) ≈ ℓ_{2}^{f} if and only if X is a strongly countabledimensional, locally compact, locally connected, separable metrizable space with no compact components. We also consider the subspace Bdd_{W} (X) ⊂ Cld_{W} (X) consisting of all non-empty bounded closed sets. In [16], it is proved that if X is an infinite-dimensional Banach space with weight w(X) then

Fin_{AW} (X) ≈ ℓ_{2}(w(X)) × ℓ_{2}^{f} and Bdd_{AW} (X) ≈ ℓ_{2}(2^{w(X)}) × ℓ_{2}^{f}.

The following is also shown in this paper:

Theorem II. If X is an infinite-dimensional separable Banach space, then

Fin_{W} (X) ≈ Bdd_{W} (X) ≈ ℓ_{2} × ℓ_{2}^{f}.

It is said that Y ⊂ X is homotopy dense in X if there is a homotopy h : X × I → X such that h_{0} = id and h_{t}(X) ⊂ Y for every t > 0. It is well-known that every homotopy dense set in an AR (resp. an ANR) is also an AR (resp. an ANR).^{2} In the way of the proof of Theorem II, we show the following:

Theorem III. Let X be a separable metric space. If the complement of any finite union of open balls in X has only finitely many path-components all of which are closed and unbounded, then Cld_{W} (X) is an AR and Fin_{W} (X) is homotopy dense in Cld_{W} (X). In particular, ℌ is an AR if Fin_{W} (X) ⊂ ℌ ⊂ Cld_{W} (X).

From Theorem III above, it follows that Cld_{W} (X) is an AR and Fin_{W} (X) is homotopy dense in Cld_{W} (X) for every infinite-dimensional separable Banach space X. Thus, we have an alternative proof of the result in [11]. To prove Theorem III, we give a condition on a metrizable Lawson semilattice X and its subsemilattice Y under which X is an ANR and Y is homotopy dense in X (Theorem 5.1). See §5 for the definition of Lawson semilattice.

Furthermore, we prove the following:

1It is said that X is strongly countable-dimensional if X is a countable union of finitedimensional closed subsets.

2This fact follows from [9, Chapter IV, Theorem 6.3].

Theorem IV. Let X be a separable metric space. If the complement of any finite union of open balls in X has only finitely many path-components, all of which are closed in X, then Fin_{W} (X) and Cld_{W} (X) are ANR’s.

For a closed subspace Y ⊂ X, we have Cld(Y) ⊂ Cld(X) as sets. Due to [2, Proposition 2.1], the Attouch-Wets topology on Cld(Y) coincides with the subspace topologies of the Attouch-Wets topology on Cld(X) as well as the Vietoris and the Fell topologies. However, the Wijsman topology on Cld(Y) does not necessarily coincide with the subspace topology inherited from the Wijsman topology on Cld(X) (see Example in Preliminaries). We also discuss the space Cld(Y) with the subspace topology inherited from Cld_{W} (X), which is called the relative Wijsman topology. Theorem IV above is proved in this setting.

2. Preliminaries

Let X = (X, d) be a metric space. The open ball and the closed ball centered at x with radius ε are denoted by B(x, ε) and \overline{B}(x, ε), respectively. By C(X), we denote the set of all continuous real-valued functions on X. In this paper, by a ‘map’, we mean a continuous function. By identifying each A ∈ Cld(X) with the map X ∋ x ↦ d(x, A) ∈ ℝ, we can regard Cld(X) ⊂ C(X), whence Cld(X) has various topologies inherited from C(X). The Wijsman topology on Cld(X) is the topology of point-wise convergence, which depends on the metric d for X. For each x ∈ X and r > 0, we define

U^{−}(x, r) = {A ∈ Cld(X) | d(x, A) < r};
U^{+}(x, r) = {A ∈ Cld(X) | d(x, A) > r}.^{3}

It is easy to see that these are open sets in Cld_{W} (X) which form an open subbasis for Cld_{W} (X). Moreover, to generate the Wijsman topology, it suffices to take points x in a dense subset of X.

For each k ∈ ℕ, we denote

Fin^{k}(X) = {A ∈ Fin(X) | card A ⩽ k}.

It is easily observed that X is homeomorphic to the subspace Fin_{W}^{1} (X) of Cld_{W} (X). Then, we can regard X = Fin_{W}^{1} (X) ⊂ Cld_{W} (X).

It is well-known that Cld_{W} (X) is metrizable if and only if X is separable, whence we can define an admissible metric d_{W} by using a countable dense set {x_{i} |∈ i ∈ ℕ} in X as follows:

d_{W} (A, B) = sup_{i∈ℕ}min{2^{−i}, |d(x_{i}, A) − d(x_{i}, B)|}.

3Although d(x, A) < r ⇔ B(x, r) ∩ A ≠ ∅ and d(x, A) > r ⇒ \overline{B}(x, r) ∩ A = ∅, it should be noticed that \overline{B}(x, r)∩A = ∅ ⇏ d(x, A) > r. In fact, let A = ⋃_{n∈ℕ}[1/n, ∞)e_{n} ∈ Cld(ℓ_{2}), where \{e_{n} | n ∈ ℕ\} is the canonical orthonormal basis for ℓ_{2}. Then, \overline{B}(0, 1) ∩ A = ∅ but d(0, A) = 1.

If d is complete then d_{W} is complete [4, Theorem 2.5.4].^{4} Thus, the space Cld_{W} (X) is completely metrizable for every separable complete metric space X.

For a closed subspace Y ⊂ X, the Wijsman topology on Cld(Y) is defined by using the metric d_{Y} = d|Y^{2} inherited from X, and the space Cld_{W} (Y) admits this topology. On the other hand, as mentioned in Introduction, Cld(Y) has the subspace topology inherited from Cld_{W} (X), called the relative Wijsman topology. The following example shows that Fin_{W}^{2} (Y) is not the subspace of Fin_{W}^{2} (X).

Example. Let X = ℓ_{2} be the Hilbert space, Y = {x ∈ ℓ_{2} | ∥x∥ ⩽ 1} the unit closed ball of X and {e_{n} | n ∈ ℕ} the standard orthonormal base of X. Fix δ > 0 and define

a_{n} = \frac{1}{\sqrt{1 + δ^{2}}} (e_{n+1} + δe_{1}), n ∈ ℕ.

Then, we have A_{n} = {0, a_{n}} ∈ Fin^{2}(Y). To see that A_{n} ↛ {0} in Cld_{W} (X), consider v = te_{1} ∈ X (= ℓ_{2}), where

t > \frac{\sqrt{1 + δ^{2}}}{2δ} .

For each n ∈ ℕ, we have

∥v − a_{n}∥^{2} = ∥(t − \frac{δ}{\sqrt{1 + δ^{2}}} ) e_{1} − \frac{1}{\sqrt{1 + δ^{2}}} e_{n+1}∥^{2}

= t^{2} − \frac{2tδ}{\sqrt{1 + δ^{2}}} + 1 < t^{2} = ∥v∥^{2},

hence d(v, A_{n}) = ∥v − a_{n}∥ ↛ ∥v∥ = d(v, {0}).

Now, we should find δ > 0 such that A_{n} → {0} in Cld_{W} (Y). In order to compute the upper estimate for δ, consider

x = (x_{1}, …, x_{k}, 0, 0, …) ∈ Y ∩ ℓ_{2}^{f} ⊂ X = ℓ_{2}.

For each n ⩾ k,

∥x − a_{n}∥^{2} = (x_{1} − \frac{δ}{\sqrt{1 + δ^{2}}} )^{2} + ∑_{i=2}^{k} x_{i}^{2} + \frac{1}{1 + δ^{2}}

and ∥x∥^{2} = ∑_{i=1}^{k} x_{i}^{2}.

d_{W} (A, B) = ∑_{i∈ℕ}2^{−i} min{1, |d(x_{i}, A) − d(x_{i}, B)|}.

4In [4], the following metric is used but it is uniformly equivalent to ours:

Then, d(x, A_{n}) → d(x, {0}) if and only if ∥x − a_{n}∥^{2} ⩾ ∥x∥^{2} for n ⩾ k. Since |x_{1}| ⩽ ∥x∥ ⩽ 1, it follows that

∥x − a_{n}∥^{2} − ∥x∥^{2} = (x_{1} − \frac{δ}{\sqrt{1 + δ^{2}}} )^{2} + \frac{1}{1 + δ^{2}} − x_{1}^{2}

= 1 − \frac{2x_{1}δ}{\sqrt{1 + δ^{2}}} ⩾ 1 − \frac{2δ}{\sqrt{1 + δ^{2}}} .

Choose 0 < δ ⩽ 1/\sqrt{3}. Then, ∥x − a_{n}∥^{2} ⩾ ∥x∥^{2}, hence d(x, A_{n}) → d(x, \{0\}) for every x ∈ Y ∩ ℓ_{2}^{f}. Since Y ∩ ℓ_{2}^{f} is dense in Y, it follows that A_{n} → {0} in Cld_{W} (Y).

To prove Theorems I and II, we need characterizations of ℓ_{2} and ℓ_{2} × ℓ_{2}^{f}. The following characterization of ℓ_{2} is due to Toruńczyk [19] (cf. [20]):

Theorem 2.1. In order that X ≈ ℓ_{2}, it is necessary and sufficient that X is a separable completely metrizable AR which has the discrete approximation property, that is, each map f : ⨁_{n∈ℕ} I^{n} → X is approximated by maps g : ⨁_{n∈ℕ} I^{n} → X such that {g(I^{n}) | n ∈ ℕ} is discrete in X.

To state the characterization of ℓ_{2} × ℓ_{2}^{f} due to Bestvina and Mogilski [5], we need some notions. A metrizable space X is σ-completely metrizable if X is a countable union of completely metrizable closed subsets. A closed set A ⊂ X is a (strong) Z-set in X if there are maps f : X → X \ A arbitrarily close to id (such that A ∩ cl f(X) = ∅). A countable union of (strong) Z-sets is called a (strong) Z_{σ}-set. When X itself is a (strong) Z_{σ}-set in X, we call X a (strong) Z_{σ}-space. For a class C of spaces, X is strongly universal for C if given a map f : A → X of A ∈ C such that f|B is a Zembedding of a closed set B ⊂ A, there exist Z-embeddings g : A → X such that g|B = f|B and which are arbitrarily close to f. In these definitions, the phrase ‘arbitrarily close’ is understood with respect to the limitation topology. In case X = (X, d) is a metric space, given a collection M of maps from a space Y to X, a map f : Y → X is arbitrarily close to maps in M if for each α : X → (0, 1) there is g ∈ M such that d(f(y), g(y)) < α(f(y)) for every y ∈ Y. The following is Corollary 6.3 in [5].

Theorem 2.2. In order that X ≈ ℓ_{2} ×ℓ_{2}^{f}, it is necessary and sufficient that X is a separable σ-completely metrizable AR which is a strong Z_{σ}-space and it is strongly universal for separable completely metrizable spaces.

In case X is a homotopy dense set in ℓ_{2}, X has the discrete approximation property (cf. [3, Theorem 1.3.2] or [1]), hence every Z-set in X is a strong Z-set by Proposition 1.7 in [5]. Then, we have the following:

Theorem 2.3. For a homotopy dense set X in ℓ_{2}, if X is a σ-completely metrizable Z_{σ}-space and it is strongly universal for separable completely metrizable spaces, then X ≈ ℓ_{2} × ℓ_{2}^{f}.

3. Proof of Theorem I

It should be remarked that Cld_{W} (X) is separable (completely) metrizable if and only if X is separable (completely) metrizable. In [11], it has been proved that Cld_{W} (X) is an AR for an infinite-dimensional separable Banach space X. Thus, it remains to verify the discrete approximation property.

Let X = (X, ∥ · ∥) be a normed linear space and d the metric induced by the norm ∥ · ∥ (i.e., d(x, y) = ∥x − y∥). By B_{X} and S_{X}, we denote the unit closed ball and the unit sphere of X, respectively. For Cld_{W} (X), the metric d_{W} is defined by a countable dense set {x_{i} | i ∈ ℕ} in X, where x_{1} = 0.

To prove Theorem I, we need the following lemma. Since it will be also used in the proof of Theorem II, it is formulated in a general setting.

Lemma 3.1. Let ℌ be a subspace of Cld_{W} (X) and W an open set in ℌ. For each map α : ℌ → (0, 1), there exists a map γ : W → (0, ∞) such that

(*) A ∈ W, A′ ∈ ℌ, A ∩ γ(A)B_{X} = A′ ∩ γ(A)B_{X}

⇒ A′ ∈ W, d_{W} (A, A′) < α(A).

Proof. For each A ∈ W, we define

i(A) = min{i ∈ ℕ | 2^{−i} < α(A) and 2^{−i} < d_{W} (A, ℌ \ W)} and
r(A) = max{∥x_{i}∥ | i ⩽ i(A)}.

Then, r : ℌ → [0, ∞) is upper semi-continuous. Indeed, let A ∈ ℌ and t > 0 such that r(A) < t. Since α is continuous, we can choose δ > 0 so that if A′ ∈ ℌ and d_{W} (A, A′) < δ then 2^{−i(A)} < α(A′), whence i(A′) ⩽ i(A) by the definition of i(A′), which implies that r(A′) ⩽ r(A) < t by the definition of r(A′). Thus, there exists a map γ : ℌ → (0, ∞) such that

γ(A) > 3 max{r(A), d(0, A)} for each A ∈ ℌ.

To see that γ satisfies condition (*), suppose

A ∈ W, A′ ∈ ℌ and A ∩ γ(A)B_{X} = A′ ∩ γ(A)B_{X}.

Note that A ∩ γ(A)B_{X} ≠ ∅ because d(0, A) < \frac{1}{3}γ(A). For each i ⩽ i(A), since ∥x_{i}∥ ⩽ r(A) < \frac{1}{3}γ(A), it follows that

d(x_{i}, A) = d(x_{i}, A ∩ γ(A)B_{X})
= d(x_{i}, A′ ∩ γ(A)B_{X}) = d(x_{i}, A′).

Then, d_{W} (A, A′) < 2^{−i(A)} < α(A). Since 2^{−i(A)} < d_{W} (A, ℌ \ W), we have also A′ ∈ W.

Theorem 3.2. For every infinite-dimensional separable Banach space X, Cld_{W} (X) has the discrete approximation property. Hence, Cld_{W} (X) ≈ ℓ_{2}.

Proof. For each map α : Cld_{W} (X) → (0, 1), let γ : Cld_{W} (X) → (0, ∞) be the map obtained by Lemma 3.1. On the other hand, S_{X} has a countableinfinite \frac{1}{2}-discrete set \{e_{n} | n ∈ ℕ\} (cf. [2]). Taking v ∈ S_{X}, we define f : Cld_{W} (X) × ℕ → Cld_{W} (X) as follows:

f(A, n) = (A ∩ γ(A)B_{X}) ∪ γ(A)S_{X} ∪ {(γ(A) + 2)v + e_{n}}.

Then, d_{W} (f(A, n), A) < α(A) for every A ∈ Cld_{W} (X) and n ∈ ℕ, by (*) in Lemma 3.1.

To verify the continuity of f, observe that

d(x, f(A, n))

= \begin{cases}min\{γ(A) − ∥x∥, d(x, A)\} & if ∥x∥ ⩽ γ(A), \\ min\{∥x∥ − γ(A), ∥(γ(A) + 2)v + e_{n} − x∥\} & if ∥x∥ ⩾ γ(A). \end{cases}

For each x ∈ X, (A, n) ↦ d(x, f(A, n)) is continuous, which implies that f is continuous.

It remains to show that

{f(Cld_{W} (X) × {n}) | n ∈ ℕ}

is discrete in Cld_{W} (X). On the contrary, assume that there exist A_{j} ∈ Cld_{W} (X), j ∈ ℕ, and n_{1} < n_{2} < ⋯ ∈ ℕ such that (f(A_{j}, n_{j}))_{j∈ℕ} converges to A ∈ Cld_{W} (X).

In case sup_{j∈ℕ} γ(A_{j}) = ∞, we may assume that γ(A_{j}) → ∞ as j → ∞. Observe that d_{W} (f(A_{j}, n_{j}), A_{j}) → 0. Then, A_{j} → A, hence γ(A_{j}) → γ(A) by the continuity of γ. This is a contradiction.

When sup_{j∈ℕ} γ(A_{j}) < ∞, it can be assumed that lim_{j→∞} γ(A_{j}) = r. We show that A ⊂ rB_{X}. Suppose that ∥a∥ > r for some a ∈ A and choose i_{0} ⩽ j_{0} ∈ ℕ so that d(x_{i_{0}}, a) < s/6 and

j ⩾ j_{0} ⇒ |γ(A_{j}) − r| < s/2, d_{W} (f(A_{j}, n_{j}), A) < min{s/6, 2^{−i_{0}}},

where s = min{∥a∥ − r, 1/4}. Then, for every j ⩾ j_{0},

d(a, f(A_{j}, n_{j})) ⩽ d(a, x_{i_{0}}) + d(x_{i_{0}}, A) + |d(x_{i_{0}}, f(A_{j}, n_{j})) − d(x_{i_{0}}, A)|
< s/6 + s/6 + s/6 = s/2.

On the other hand,

d(a, γ(A_{j})B_{X}) ⩾ ∥a∥ − γ(A_{j}) > ∥a∥ − r − s/2 ⩾ s/2.

Therefore, d(a, (γ(A_{j}) + 2)v + e_{n_{j}}) < s/2. Then, we have

d(a, (r + 2)v + e_{n_{j}}) ⩽ d(a, (γ(A_{j}) + 2)v + e_{n_{j}})
+ d((γ(A_{j}) + 2)v + e_{n_{j}}, (r + 2)v + e_{n_{j}})
< s/2 + |γ(A_{j}) − r| < s.

Then, for j ≠ j′ ⩾ j_{0},

1/2 < d(e_{n_{j}}, e_{n_{j′}}) = d((r + 2)v + e_{n_{j}}, (r + 2)v + e_{n_{j′}})
⩽ d(a, (r + 2)v + e_{n_{j}}) + d(a, (r + 2)v + e_{n_{j′}}) < 2s ⩽ 1/2,

which is a contradiction. Therefore, A ⊂ rB_{X}.

Now, choose i_{1} ∈ ℕ so that d(x_{i_{1}}, (r + 2)v) < 1/4, whence

d(x_{i_{1}}, A) ⩾ d(x_{i_{1}}, rB_{X})
⩾ d((r + 2)v, rB_{X}) − d(x_{i_{1}}, (r + 2)v)
> 2 − 1/4 = 7/4.

Moreover, choose j_{1} ∈ ℕ so that if j ⩾ j_{1} then |γ(A_{j_{1}}) − r| < 1/2, whence

d(x_{i_{1}}, f(A_{j}, n_{j})) ⩽ d(x_{i_{1}}, (γ(A_{j}) + 2)v)
⩽ d(x_{i_{1}}, (r + 2)v) + d((r + 2)v, (γ(A_{j}) + 2)v)
< 1/4 + |γ(A_{j}) − r| < 3/4.

Then, for every j ⩾ j_{1},

d_{W} (f(A_{j}, n_{j})), A) ⩾ min{2^{−i_{1}}, |d(x_{i_{1}}, f(A_{j}, n_{j})) − d(x_{i_{1}}, A)|}
⩾ min{2^{−i_{1}}, 7/4 − 3/4} = 2^{−i_{1}},

which contradicts to f(A_{j}, n_{j}) → A. Consequently,

{f(Cld_{W} (X) × {n}) | n ∈ ℕ}

is discrete in Cld_{W} (X).

4. Proof of Theorem II for Bdd_{W} (X)

By Theorem I, we can apply Theorem 2.3 to prove that Bdd_{W} (X) ≈ ℓ_{2} × ℓ_{2}^{f}. We first show the following:

Lemma 4.1. For every separable normed linear space X, Bdd_{W} (X) is homotopy dense in Cld_{W} (X).

Proof. We define θ : Cld_{W} (X) × I → Cld_{W} (X) by θ_{0} = id and

θ(A, t) = (A ∩ t^{−1}B_{X}) ∪ t^{−1}S_{X}.^{5}

To verify the continuity of θ, observe that

d(x, θ(A, t)) = \begin{cases}min\{t^{−1} − ∥x∥, d(x, A)\} & if ∥x∥ ⩽ t^{−1}, \\ ∥x∥ − t^{−1} & if ∥x∥ > t^{−1}. \end{cases}

For each x ∈ X, (A, t) ↦ d(x, θ(A, t)) is continuous, which implies that θ is continuous.

Notice Bdd_{W} (X) = ⋃_{k∈ℕ} Cld(kB_{X}).

Lemma 4.2. For every separable normed linear space X, each Cld(kB_{X}) is closed in Cld_{W} (X). Thus, if X is a separable Banach space, then each Cld(kB_{X}) is completely metrizable, hence Bdd_{W} (X) is σ-completely metrizable.

5As compared with the contraction θ in [2], the parameter is different. Here, it is not necessary that θ_{1} is the constant.

Proof. For each A ∈ Cld_{W} (X) \ Cld(kB_{X}), we have a ∈ A \ kB_{X}. Let r = ∥a∥ − k = d(a, kB_{X}) > 0. Then,

A ∈ U^{−}(a, r) ⊂ Cld_{W} (X) \ Cld(kB_{X}).

Therefore, Cld_{W} (X) \ Cld(kB_{X}) is open in Cld_{W} (X), that is, Cld(kB_{X}) is closed in Cld_{W} (X).

Lemma 4.3. For every separable normed linear space X, each Cld(kB_{X}) is a Z-set in Bdd_{W} (X), hence Bdd_{W} (X) is a Z_{σ}-space.

Proof. Let θ : Cld_{W} (X) × I → Cld_{W} (X) be the homotopy defined in the proof of Lemma 4.1. Then, θ(Bdd_{W} (X) × I) ⊂ Bdd_{W} (X). For each map α : Bdd_{W} (X) → (0, 1), define ξ : Bdd_{W} (X) → (0, 1) as follows:

ξ(A) = sup{t > 0 | diam_{d_{W}} θ({A} × [0, t]) < α(A)}.

Then, ξ is lower semi-continuous. Indeed, if ξ(A) > s then we have s < s′ < ξ(A) and

ε = α(A) − diam_{d_{W}} θ({A} × [0, s′]) > 0.

By the continuity of θ and α, we have δ > 0 such that if A′ ∈ Fin_{W} (X) and d_{W} (A, A′) < δ then |α(A) − α(A′)| < \frac{1}{3}ε and d_{W} (θ(A, t), θ(A′, t)) < \frac{1}{3}ε for all t ∈ [0, s′], whence

diam_{d_{W}} θ(\{A′\} × [0, s′]) < diam_{d_{W}} θ(\{A\} × [0, s′]) + \frac{2}{3}ε
= α(A) − \frac{1}{3}ε < α(A′),

which means that ξ(A′) ⩾ s′ > s.

Thus, we have a map f : Bdd_{W} (X) → Bdd_{W} (X) defined by

f(A) = θ(A, β(A)) = A ∪ β(A)^{−1}S_{X},

where β : Bdd_{W} (X) → (0, 1) is a map such that

β(A) < min{(k + 1)^{−1}, ξ(A)}.

Observe that d_{W} (f(A), A) < α(A) and β(A)^{−1} > k + 1 for each A ∈ Bdd_{W} (X). Then, it follows that Cld(kB_{X}) ∩ f(Bdd_{W} (X)) = ∅. Thus, Cld(kB_{X}) is a Z-set in Bdd_{W} (X).

It remains to prove the strong universality of Bdd_{W} (X). To this end, we use the following fact:

Proposition 4.4. For every infinite-dimensional separable Banach space X, the unit sphere S_{X} is homeomorphic to ℓ_{2}.

Proof. This is well-known but we give a proof for readers’ convenience.

First note that X ≈ ℓ_{2} (cf. [19, §6]). As an open set in X ≈ ℓ_{2}, (1, 2)S_{X} is an ℓ_{2}-manifold. It is easy to see that S_{X} ∪ 2S_{X} is a strong Z-set in [1, 2]S_{X}. Then, [1, 2]S_{X} is an ℓ_{2}-manifold by [20, Theorem B1] (cf. [18, Theorem 5.2]). By using [19, Theorem 4.1], we can see that S_{X} is also an ℓ_{2}-manifold. On the other hand, S_{X} is contractible because S_{X} is a retract of X \ {0} and X \ {0} ≈ ℓ_{2} \ {0} is homotopically trivial. Therefore, S_{X} ≈ ℓ_{2}.

Theorem 4.5. For every infinite-dimensional separable Banach space X, Bdd_{W} (X) is strongly universal for separable completely metrizable spaces. Consequently, Bdd_{W} (X) ≈ ℓ_{2} × ℓ_{2}^{f}.

Proof. First of all, note that the class of separable completely metrizable spaces is hereditary with respect to both closed subsets and open subsets. Due to the remark before Theorem 2.3, it follows from Lemma 4.1 that every Z-set in Bdd_{W} (X) is a strong Z-set. Then, by Proposition 2.2 in [5], it suffices to prove that each open set W ⊂ Bdd_{W} (X) is universal for separable completely metrizable spaces, that is, for every separable completely metrizable space Y, each map f : Y → W can be approximated by Z-embeddings.

For each map α : W → (0, 1), we apply Lemma 3.1 to obtain a map β : W → (0, 1) such that

(*) A ∈ W, A′ ∈ Bdd(X), A ∩ β(A)^{−1}B_{X} = A′ ∩ β(A)^{−1}B_{X}

⇒ A′ ∈ W, d_{W} (A, A′) < α(A).

Since the Hilbert space ℓ_{2} is universal for separable completely metrizable spaces, we have a closed embedding h : Y → S_{X} by Proposition 4.4. By taking any v ∈ S_{X} and using the homotopy θ : Cld_{W} (X) × I → Cld_{W} (X) defined in the proof of Lemma 4.1, we can define a map g : Y → W by

g(y) = θ(f(y), β(f(y))) ∪ {(β(f(y))^{−1} + 2)v + h(y)}.

Then, it follows from (*) that d_{W} (f(y), g(y)) < α(f(y)) for every y ∈ Y. As is easily observed, g is injective.

To see that g is a closed embedding, let y_{i} ∈ Y, i ∈ ℕ, such that g(y_{i}) → A ∈ W. Then, b = inf_{i∈ℕ} β(f(y_{i})) > 0. Otherwise, by taking a subsequence, it can be assumed that β(f(y_{i})) → 0, that is, β(f(y_{i}))^{−1} → ∞, whence d_{W} (f(y_{i}), g(y_{i})) → 0, so f(y_{i}) → A. Hence, β(f(y_{i})) → β(A) ≠ 0, which is a contradiction. Furthermore, A ∩ ((b^{−1} + 2)v + \frac{3}{2}B_{X}) ≠ ∅. In fact, if A ∈ U^{+}((b^{−1} + 2)v, \frac{3}{2}) then g(y_{i}) ∈ U^{+}((b^{−1} + 2)v, \frac{3}{2}) for sufficiently large i ∈ ℕ. On the other hand, for sufficiently large i ∈ ℕ, |β(f(y_{i}))^{−1}−b^{−1}| < \frac{1}{2}, whence

∥(β(f(y_{i}))^{−1} + 2)v + h(y_{i}) − (b^{−1} + 2)v∥ ⩽ |β(f(y_{i}))^{−1} − b^{−1}| + 1 < \frac{3}{2}.

This is a contradiction.

Now, let c ∈ A ∩ ((b^{−1} + 2)v + \frac{3}{2}B_{X}). For each 0 < ε < \frac{1}{4}, we can choose i_{0} ∈ ℕ so that if i ⩾ i_{0} then g(y_{i}) ∈ U^{−}(c, ε) and |β(f(y_{i}))^{−1} −b^{−1}| < ε < \frac{1}{4}, whence

∥(β(f(y_{i}))^{−1} + 2)v + h(y_{i}) − c∥ = ∥g(y_{i}) − c∥ < ε

because ∥c∥ ⩾ b^{−1}+ \frac{1}{2} > β(f(y_{i}))^{−1}+ \frac{1}{4}. Thus, (β(f(y_{i}))^{−1}+2)v+h(y_{i}) → c. Since β(f(y_{i}))^{−1} → b^{−1}, it follows that h(y_{i}) → c − (b^{−1} + 2)v. Therefore, (y_{i})_{i∈ℕ} is convergent in Y. Hence, g is a closed embedding.

By Lemma 4.3, g(Y) is a Z_{σ}-set in W. Since g(Y) ≈ Y is completely metrizable, g(Y) is a Z-set in W by [6, Lemma 2.4]. Thus, g : Y → W is a Z-embedding. The proof is completed.

5. Homotopy Dense Subsemilattices of Lawson Semilattices

A topological semilattice is a topological space S equipped with a continuous operator ∨ : S ×S → S which is reflexive, commutative and associative (i.e., x ∨ x = x, x ∨ y = y ∨ x, (x ∨ y) ∨ z = x ∨ (y ∨ z)). A topological semilattice S is called a Lawson semilattice if S admits an open basis consisting of subsemilattices [12].

In [2], it is shown that a metrizable Lawson semilattice X is an ANR (resp. an AR) if and only if it is locally path-connected (resp. connected and locally path-connected). Here, we introduce a relative version of local path-connectedness. A subset Y ⊂ X is relatively LC^{0} in X if for every x ∈ X, each neighborhood U of x in X contains a smaller neighborhood V of x such that every two points of V ∩ Y can be joined by a path in U ∩ X. In this section, we shall prove the following theorem.

Theorem 5.1. Let X be a metrizable Lawson semilattice with Y ⊂ X a dense subsemilattice. If Y is relatively LC^{0} in X (and Y is path-connected), then X is an ANR (an AR) and Y is homotopy dense in X, hence Y is also an ANR (an AR).

To prove Theorem 5.1 above, we will use the following result in [15]:

Theorem 5.2. Let X be a metric space with Y ⊂ X a dense set. Assume that there exist a zero-sequence U = {U_{n} | n ∈ ℕ} of open covers of X and a map f : |T N(U)| → Y such that f(U) ∈ U for U ∈ ⋃_{n∈ℕ} U_{n} = T N(U)^{(0)} and

lim_{n→∞} mesh{f(σ) | σ ∈ N(U_{n} ∪ U_{n+1})} = 0.

Then, X is an ANR and Y is homotopy dense in X.

Here, U is called a zero-sequence if lim_{n→∞} mesh U_{n} = 0, and T N(U) is a simplicial complex defined as the union ⋃_{n∈ℕ} N(U_{n} ∪ U_{n+1}) of the nerves of the covers U_{n} ∪ U_{n+1}, where we regard U_{n} ∩ U_{n+1} = ∅.

Lemma 5.3. Every Lawson semilattice X is k-aspherical for any k > 0, that is, each map f : S^{k} → X extends to f̃: B^{k+1} → X.

Proof. Identify S^{k} = Fin_{V}^{1} (S^{k}) ⊂ Fin_{V} (S^{k}) and X = Fin_{V}^{1} (X) ⊂ Fin_{V} (X). Then, f has the extension f_{V} : Fin_{V} (S^{k}) → Fin_{V} (X). Since Fin_{V} (X) is a free Lawson semilattice over X, there exists a retraction r : Fin_{V} (X) → X (see [2]). By [7], we have φ : B^{k+1} → Fin_{V}^{3} (S^{k}) with φ|S^{k} = id. Then, f̃ = rf_{V} φ : B^{k+1} → X is an extension of f.

Proof of Theorem 5.1. Fix an admissible metric d on X. For each n ∈ ℕ, let V_{n} be an open cover of X such that mesh V_{n} < 2^{−n} and each V ∈ V_{n} is a subsemilattice of X. Since Y is relatively LC^{0} in X, each V_{n} has an open refinement U_{n} such that each U ∈ U_{n} is contained in some V ∈ V_{n} so that every pair of points in U ∩ Y can be connected by a path in V.

Now, for each n ∈ ℕ, we can define f_{n} : |N(U_{n} ∪ U_{n+1})| → Y such that f_{n}(U) ∈ U for U ∈ U_{n} ∪ U_{n+1} and for every simplex σ ∈ N(U_{n} ∪ U_{n+1}) the image f_{n}(σ) is contained in a member of V_{n}. This is possible because of Lemma 5.3. Each f_{n} can be inductively defined in such a way that f_{n}|N(U_{n+1}) = f_{n+1}|N(U_{n+1}). Then, we have the map f : |T N(U)| → Y defined by f||N(U_{n} ∪ U_{n+1})| = f_{n}, which satisfies the assumptions of Theorem 5.2, hence X is an ANR and Y is homotopy dense in X.

Moreover, if X or Y is path-connected, then it is n-connected for every n ∈ ℕ by Lemma 5.3. Then, X is an AR.

The case X = Y in Theorem 5.1 is [2, Proposition 3.2], that is,

Corollary 5.4. Let X be a metrizable Lawson semilattice. Then, X is an ANR (an AR) if and only if X is locally path-connected (and pathconnected).

For an arbitrary subset Y ⊂ X, we have Fin(Y) ⊂ Comp(Y) ⊂ Cld(X). It should be noticed that Cld(Y) ⊄ Cld(X) unless Y is closed in X. As saw in Preliminaries, Fin_{W} (Y) (nor Cld_{W} (Y)) is not a subspace of Cld_{W} (X) even if Y is closed in X.

When Y is dense in X, we can use a countable dense subset of Y to define the metric d_{W} for Cld_{W} (X). Then, Comp_{W} (Y) and Fin_{W} (Y) are subspaces of Cld_{W} (X). Moreover, we have the following:

Proposition 5.5. Let X be separable metric space and Y a dense subspace of X with the metric inherited from X. Then, the space Cld_{W} (Y) can be naturally embedded in Cld_{W} (X) by the closure operator cl_{X} : Cld_{W} (Y) → Cld_{W} (X). Namely, the space Cld_{W} (Y) can be identified with the subspace {A ∈ Cld(X) | cl_{X}(A ∩ Y) = A} of Cld_{W} (X).

Proposition 5.6. For a dense subset Y of a separable metric space X, Fin_{W} (Y) is a dense semilattice of Cld_{W} (X).

Proof. Let A ∈ Cld_{W} (X). For each neighborhood W of A in Cld_{W} (X), we have p_{1}, …, p_{n} ∈ X, r_{1}, …, r_{n} > 0 and m ⩽ n such that

A ∈ ⋂_{i=1}^{m} U^{−}(p_{i}, r_{i}) ∩ ⋂_{j=m+1}^{n} U^{+}(p_{j}, r_{j}) ⊂ W.

For each 1 ⩽ i ⩽ m, B(p_{i}, r_{i}) \ ⋃_{j=m+1}^{n} \overline{B}(p_{j}, r_{j}) ≠ ∅ because it contains a point of A. Since Y is dense in X, we have

y_{i} ∈ Y ∩ B(p_{i}, r_{i}) \ ⋃_{j=m+1}^{n} \overline{B}(p_{j}, r_{j}).

Then, A′ = {a_{1}, …, a_{m}} ∈ Fin_{W} (Y) ∩ W. Therefore, Fin_{W} (Y) is dense in Cld_{W} (X).

By applying Theorem 5.1 in the above setting, we have the following:

Corollary 5.7. Let X be a separable metric space with Y ⊂ X a dense subset. If Fin_{W} (Y) is relatively LC^{0} in Cld_{W} (X) (and Fin_{W} (Y) is pathconnected), then Cld_{W} (X) is an ANR (an AR) and Fin_{W} (Y) is homotopy dense in Cld_{W} (X), hence ℌ is an ANR (an AR) if Fin_{W} (Y) ⊂ ℌ ⊂ Cld_{W} (Y).

6. Proof of Theorem III

In this section, we prove Theorem III. Here the metric for a metric space is denoted by d. To apply Theorem 5.1, we first show the following:

Proposition 6.1. For an arbitrary metric space X, Cld_{W} (X) is a Lawson semilattice with respect to the union operator ∪.

Proof. For each x ∈ X,

(A, B) ↦ d(x, A ∪ B) = min{d(x, A), d(x, B)}

is continuous, which implies that the union operator ∪ is continuous.

Observe that U^{−}(x, r) and U^{+}(x, r) are subsemilattices of Cld_{W} (X) for each x ∈ X and r > 0. Since the intersection of subsemilattices is also a subsemilattice, Cld_{W} (X) has an open basis consisting of subsemilattices, hence Cld_{W} (X) is a Lawson semilattice.

Lemma 6.2. Let X be a separable metric space and Y a path-connected subset of X. Then, each A, B ∈ Fin_{W} (Y) can be connected by a path γ : I → Fin_{W} (Y) such that each γ(t) contains A or B.

Proof. For each a ∈ A and b ∈ B, we have a path γ_{a,b} : I → Y from a to b. Now, we define a path γ : I → Fin_{W} (Y) by

γ(t) = A ∪ ⋃{γ_{a,b}(t) | a ∈ A, b ∈ B}.

Then, γ is a path in Fin_{W} (Y) from A to A ∪ B. By the same argument, we can find a path from B to A ∪ B.

To prove the next result, we shall use a well-known combinatorial fact, called König’s Lemma: Every finitely-branching infinite tree contains an infinite branch (cf. [10, (4.12)]). A tree is a partially ordered set (T, <) such that for every x ∈ T, the set {y ∈ T | y < x} is well-ordered (in our case: finite, linearly ordered). A branch through T is a maximal linearly ordered subset of T. In the proof of the following theorem, we consider a tree of sets with the reversed inclusion as the partial order. It is said that A, B ⊂ X are strongly disjoint if

dist(A, B) = inf{d(x, y) | x ∈ A, y ∈ B} > 0.

Theorem 6.3. Let X be a separable metric space. If the complement of any finite union of open balls in X has only finitely many path-components, all of which are closed and unbounded or compact, then Cld_{W} (X) is an ANR and Fin_{W} (X) is homotopy dense in Cld_{W} (X). Hence ℌ is an ANR whenever Fin_{W} (X) ⊂ ℌ ⊂ Cld_{W} (X).

Proof. By Corollary 5.7, it suffices to show that Fin_{W} (X) is relatively LC^{0} in Cld_{W} (X). Note that X itself has only finitely many path-components (the case the number of open balls is zero). We only consider the case X is not compact, hence X is unbounded by the assumption.

Let A ∈ Cld_{W} (X) and U a neighborhood of A in Cld_{W} (X). Then,

A ∈ ⋂_{i<k} U^{−}(p_{i}, r_{i}) ∩ ⋂_{j<l} U^{+}(q_{j}, s_{j}) ⊂ U,

for some p_{i}, q_{j} ∈ X and r_{i}, s_{j} > 0. Choose ε > 0 so that A ∈ U^{+}(q_{j}, s_{j} + ε) for every j < l.

Let X_{0}, …, X_{m−1} be all path-components of X \⋃_{j<l} B(q_{j}, s_{j} +ε), where X_{i} ∩A ≠ ∅ for i < m_{0} and X_{j} ∩A = ∅ for j ⩾ m_{0}. Moreover, we can assume that X_{j} is unbounded for m_{0} ⩽ j < m_{1} and X_{j} is compact for j ⩾ m_{1}. Note that each X_{i} is not only closed but also open in X \ ⋃_{j<l} B(q_{j}, s_{j} + ε) and A is strongly disjoint from ⋃_{j<l} B(q_{j}, s_{j} + ε). For each i < k, we can choose a_{i} ∈ A ∩ B(p_{i}, r_{i}) and ε_{i} > 0 so that B(a_{i}, ε_{i}) ⊂ X_{j} ∩ B(p_{i}, r_{i}) for some j < m_{0}. For each i < m_{0}, we can choose a_{k+i} ∈ X_{i} ∩ A and ε_{k+i} > 0 so that B(a_{k+i}, ε_{k+i}) ⊂ X_{i}. Since ⋃_{m_{1}⩽j<m} X_{j} is compact, we can find t_{0}, …, t_{v−1} ∈ X and δ_{0}, …, δ_{v−1} > 0 such that

⋃_{m_{1}⩽j<m} X_{j} ⊂ ⋃_{i<v} B(t_{i}, δ_{i}) ⊂ ⋃_{i<v} B(t_{i}, 2δ_{i}) ⊂ X \ ⋃_{i<m_{0}} X_{i}.

Since A ⊂ ⋃_{i<m_{0}} X_{i}, it follows that A ∈ ⋂_{i<v} U^{+}(t_{i}, δ_{i}). Thus, A has the following neighborhood:

V = ⋂_{i<k+m_{0}} U^{−}(a_{i}, ε_{i}) ∩ ⋂_{j<l} U^{+}(q_{j}, s_{j} + ε) ∩ ⋂_{i<v} U^{+}(t_{i}, δ_{i}) ⊂ U.

Let A_{0} = {a_{i} | i < k + m_{0}} ⊂ A. Then, A_{0} ∈ V ∩ Fin_{W} (X).

We show that each B ∈ V ∩Fin_{W} (X) can be connected to A_{0} by a path in U ∩Fin_{W} (X). Let B^{∗} = B ∩⋃_{m_{0}⩽j<m_{1}} X_{j}. Applying Lemma 6.2 to A_{0} ∩X_{i} and B ∩ X_{i} for i < m_{0}, we can easily construct a path in U ∩ Fin_{W} (X) from B to A_{0} ∪ B^{∗}. For each z ∈ B^{∗}, choose m_{0} ⩽ j < m_{1} so that z ∈ X_{j}, If we can construct an infinite path f_{z} : [1, ∞) → X_{j} such that f_{z}(1) = z and lim_{t→∞} d(z, f_{z}(t)) = ∞ (whence lim_{t→∞} d(x, f_{z}(t)) = ∞ for any x ∈ X), then we can define a path ψ : I → V from A_{0} to A_{0} ∪ B^{∗} as follows: ψ(0) = A_{0} and

ψ(t) = A_{0} ∪ ⋃_{z∈B^{∗}} f_{z}(t^{−1}) for t > 0.

For any x ∈ X, since lim_{t→∞} d(x, f_{z}(t)) = ∞, d(x, ψ(t)) = d(x, A_{0}) for sufficiently small t > 0, which means that ψ is continuous at 0.

Let z ∈ B^{∗}∩X_{j} (m_{0} ⩽ j < m_{1}). Enumerate as B_{1}, B_{2}, … all open balls of the form B(x_{i}, α), where 0 < α < d(x_{i}, A) and α ∈ ℚ. By the assumption, for each n ∈ ℕ, X_{j} \ (B_{1} ∪ ⋯ ∪ B_{n}) has finitely many path-components H_{0}^{n}, …, H_{a(n)−1}^{n}. Let T = {H_{i}^{n} | n ∈ ℕ, i < a(n)}. Since X_{j} is unbounded, it follows that T is infinite. Thus, (T, ⊃) is a finitely-branching infinite tree (i.e. each element of T has only finitely many immediate successors). By König’s lemma, T contains an infinite branch X_{j} ⊃ H_{i(1)}^{1} ⊃ H_{i(2)}^{2} ⊃ · · ·. For each n ∈ ℕ, pick z_{n} ∈ H_{i(n)}^{n} and a path f_{n} : [n − 1, n] → H_{i(n−1)}^{n−1} such that f_{n}(n − 1) = z_{n−1} and f_{n}(n) = z_{n}, where H_{i(0)}^{0} = X_{j} and z_{0} = z. By joining all paths f_{1}, f_{2}, …, we can obtain a path f_{z} : [1, +∞) → Y_{j} with f_{z}(1) = z and lim_{t→∞} d(z, f_{z}(t)) = ∞.

Theorem 6.4. Let X be a separable metric space with x_{0} ∈ X. Suppose that for arbitrarily large r > 0, X \ B(x_{0}, r) has only finitely many pathcomponents, all of which are unbounded. Then, Fin_{W} (X) is path-connected.

Proof. First, note that X itself has only finitely many path-components, say X_{0}, …, X_{m−1}. For each i < m, take a_{i} ∈ U_{i}, and let A_{0} = {a_{i} | i < m} ∈ Fin_{W} (X). We show that each B ∈ Fin_{W} (X) can be connected to A_{0} by a path in Fin_{W} (X).

Each x ∈ B is contained in some X_{i(x)}, whence we have a path f_{x} : I → X_{i(x)} such that f_{x}(0) = x and f_{x}(1) = a_{i(x)}. Then, B is connected to A_{1} = {a_{i(x)} | x ∈ B} ⊂ A_{0} by a path φ : I → Fin_{W} (X) defined by φ(t) = {f_{x}(t) | x ∈ A}. Let A_{0} \ A_{1} = {a_{i} | i ∈ S}, that is,

S = {i | i ≠ i(x) for any x ∈ B}.

For i ∈ S, we will construct a path g_{i} : [0, ∞) → X_{i} such that g_{i}(0) = a_{i} and lim_{t→∞} d(a_{i}, g_{i}(t)) = ∞. Then, A_{0} and A_{1} can be connected by the path φ : I → Fin_{W} (X) defined as follows:

ψ(t) = \begin{cases}A & if t = 0, \\ \{g_{i}(t^{−1}) | i ∈ S\} ∪ A_{1} & if t > 1. \end{cases}

Thus, it follows that Fin_{W} (X) is path-connected.

The path g_{i} above can be constructed as follows: By the assumption, there are 0 < r_{1} < r_{2} < ⋯ such that lim_{n→∞} r_{n} = ∞ and each X \ B(x_{0}, r_{n}) has only finitely many path-components. Then, X_{i} \ B(x_{0}, r_{n}) has also only finitely many path-components, say H_{1}^{n}, …, H_{k(n)}^{n}. Let T = {H_{j}^{n} | n ∈ ℕ, j < k(n)}. Since X_{i} is unbounded, T is infinite. Thus, (T, ⊃) is a finitely-branching infinite tree (i.e., each element of T has only finitely many immediate successors). By König’s lemma, T contains an infinite branch X_{i} ⊃ H_{j(1)}^{1} ⊃ H_{j(2)}^{2} ⊃ · · ·. For each n ∈ ℕ, pick v_{n} ∈ H_{j(n)}^{n} and a path g_{n} : [n − 1, n] → H_{j(n−1)}^{n−1} such that f_{n}(n − 1) = v_{n−1} and f_{n}(n) = v_{n}, where H_{j(0)}^{0} = X_{j} and v_{0} = a_{i}. By joining all paths f_{1}, f_{2}, …, we can obtain a path g_{i} : [1, +∞) → X_{i} with g_{i}(1) = a_{i} and lim_{t→∞} d(a_{i}, g_{i}(t)) = ∞.

Due to Corollary 5.7, Theorem III follows from Theorems 6.3 and 6.4.

7. Proof of Theorem II for Fin_{W} (X)

As in the case of Bdd_{W} (X), we apply Theorem 2.3 to prove that Fin_{W} (X) is homeomorphic to ℓ_{2} × ℓ_{2}^{f}. In the previous section, we have shown that Fin_{W} (X) is homotopy dense in Cld_{W} (X) for an infinite-dimensional separable Banach space X.

Now, notice Fin_{W} (X) = ⋃_{k∈ℕ} Fin^{k}(X).

Lemma 7.1. For an arbitrary metric space X, each Fin^{k}(X) is closed in Cld_{W} (X). Thus, if X is complete, then each Fin^{k}(X) is completely metrizable, hence Fin_{W} (X) is σ-completely metrizable.

Proof. For each A ∈ Cld_{W} (X) \ Fin^{k}(X), choose distinct k + 1 many points a_{1}, …, a_{k+1} ∈ A and let r = \frac{1}{2} min\{d(a_{i}, a_{j}) | i ≠ j\}. Then,

A ∈ ⋂_{i=1}^{k+1} U^{−}(a_{i}, r) ⊂ Cld_{W} (X) \ Fin^{k}(X).

Therefore, Cld_{W} (X) \ Fin^{k}(X) is open in Cld_{W} (X), that is, Fin^{k}(X) is closed in Cld_{W} (X).

Lemma 7.2. For every separable normed linear space X, each Fin^{k}(X) is a Z-set in Fin_{W} (X), hence Fin_{W} (X) is a Z_{σ}-space.

Proof. By using distinct k + 1 many points v_{1}, …, v_{k+1} ∈ S_{X}, we define a homotopy ζ : Fin_{W} (X) × I → Fin_{W} (X) by ζ_{0} = id and

ζ(A, t) = A ∪ t^{−1}{v_{1}, …, v_{k+1}} for t > 0.

For each map α : Fin_{W} (X) → (0, 1), define γ : Fin_{W} (X) → (0, 1) by

γ(A) = sup{t > 0 | diam_{d_{W}} ζ({A} × [0, t]) < α(A)}.

Then, γ is lower semi-continuous. Indeed, if γ(A) > s then we have s < s′ < γ(A) and

ε = α(A) − diam_{d_{W}} ζ({A} × [0, s′]) > 0.

By the continuity of ζ and α, we have δ > 0 such that if A′ ∈ Fin_{W} (X) and d_{W} (A, A′) < δ then |α(A) − α(A′)| < \frac{1}{3}ε and d_{W} (ζ(A, t), ζ(A′, t)) < \frac{1}{3}ε for all t ∈ [0, s′], whence

diam_{d_{W}} ζ(\{A′\} × [0, s′]) < diam_{d_{W}} ζ(\{A\} × [0, s′]) + \frac{2}{3}ε
= α(A) − \frac{1}{3}ε < α(A′),

which means that γ(A′) ⩾ s′ > s.

Now, we define a map f : Fin_{W} (X) → Fin_{W} (X) as follows:

f(A) = ζ(A, β(A)) = A ∪ β(A)^{−1}{v_{1}, …, v_{k+1}},

where β : Fin_{W} (X) → (0, 1) is a map such that β(A) < γ(A). Observe that d_{W} (f(A), A) < α(A) for each A ∈ Fin_{W} (X). By the definition, Fin^{k}(X) ∩ f(Fin_{W} (X)) = ∅. Thus, we have the result.

Remark . Due to the remark before Theorem 2.3, it follows from Theorem III that every Z-set in Fin_{W} (X) is a strong Z-set. Therefore, each Fin^{k}(X) is a strong Z-set in Fin_{W} (X). However, this can be also obtained in the above proof by showing that

Fin^{k}(X) ∩ cl f(Fin_{W} (X)) = ∅.

To this end, assume that there exist A_{n} ∈ Fin_{W} (X), n ∈ ℕ, such that A = lim_{n→∞} f(A_{n}) ∈ Fin^{k}(X). Then, we will find a contradiction in the both cases inf_{n∈ℕ} β(A_{n}) = 0 and inf_{n∈ℕ} β(A_{n}) > 0.

When inf_{n∈ℕ} β(A_{n}) = 0, we may assume that lim_{n→∞} β(A_{n}) = 0. For each ε > 0, choose i_{0} ∈ ℕ so that 2^{−i_{0}} < ε. Since β(A_{n})^{−1} → ∞ as n → ∞, we have n_{0} ∈ ℕ such that

n ⩾ n_{0} ⇒ d_{W} (f(A_{n}), A) < 2^{−i_{0}−1},

β(A_{n})^{−1} > max{∥x_{i}∥ + d(x_{i}, A) + 2^{−i_{0}−1} | i = 1, …, i_{0}}.

Then, for each n ⩾ n_{0} and i ⩽ i_{0},

β(A_{n})^{−1} − ∥x_{i}∥ > d(x_{i}, A) + 2^{−i_{0}−1} > d(x_{i}, f(A_{n})),

hence d(x_{i}, f(A_{n})) = d(x_{i}, A_{n}), which implies d_{W} (f(A_{n}), A_{n}) ⩽ 2^{−i_{0}−1}. Since d_{W} (f(A_{n}), A) < 2^{−i_{0}−1} it follows that d_{W} (A_{n}, A) < 2^{−i_{0}} < ε. Thus, (A_{n})_{n∈ℕ} converges to A, hence lim_{n→∞} β(A_{n}) = β(A) > 0, which is a contradiction.

When inf_{n∈ℕ} β(A_{n}) > 0, we may assume that lim_{n→∞} β(A_{n}) = b > 0. Then, we show that z_{j} = b^{−1}v_{j} ∈ A for each j = 1, …, k + 1. If z_{j} ∉ A, choose x_{i_{j}} so that d(x_{i_{j}}, z_{i}) < \frac{1}{4}d(z_{j}, A), whence

d(x_{i_{j}}, A) ⩾ d(z_{j}, A) − d(x_{i_{j}}, z_{i}) > \frac{3}{4}d(z_{j}, A).

For sufficiently large n ∈ ℕ,

|d(x_{i_{j}}, A) − d(x_{i_{j}}, f(A_{n}))| < min\{2^{−i_{j}}, \frac{1}{4}d(z_{j}, A)\} and
|b^{−1} − β(A_{n})^{−1}| < \frac{1}{4}d(z_{j}, A),

whence it follows that

d(z_{j}, A) ⩽ d(z_{i}, x_{i_{j}}) + d(x_{i_{j}}, A)
< \frac{1}{4}d(z_{j}, A) + |d(x_{i_{j}}, A) − d(x_{i_{j}}, A_{n})| + d(x_{i_{j}}, A_{n})
< \frac{1}{2}d(z_{j}, A) + d(x_{i_{j}}, z_{i}) + d(z_{i}, A_{n})
< \frac{3}{4}d(z_{j}, A) + |b^{−1} − β(A_{n})^{−1}| < d(z_{j}, A).

This is a contradiction.

Theorem 7.3. For every infinite-dimensional separable Banach space X, Fin_{W} (X) is strongly universal for separable completely metrizable spaces. Consequently, Fin_{W} (X) ≈ ℓ_{2} × ℓ_{2}^{f}.

Proof. By the same reason as in Theorem 4.5, it suffices to prove that each open set W ⊂ Fin_{W} (X) is universal for separable completely metrizable spaces. Let Y be a separable completely metrizable space, and f : Y → W a map. For each map α : W → (0, 1), similarly to Theorem 4.5, we apply Lemma 3.1 to obtain a map β : Fin_{W} (X) → (0, 1) such that

(*) A ∈ W, A′ ∈ Fin(X), A ∩ β(A)^{−1}B_{X} = A′ ∩ β(A)^{−1}B_{X}

⇒ A′ ∈ W, d_{W} (A, A′) < α(A).

Let β_{0} = β. For n ∈ ℕ, let β_{n} : Fin_{W} (X) → I be the map defined by

β_{n}(A) = min {(n + 1)^{−1}, β(A), β(A)d_{W} (A, Fin^{n}(nB_{X}))}.

Note that n + 1 ⩽ β_{n}(A)^{−1} if β_{n}(A) ≠ 0 and β_{n}(A)^{−1} ⩽ β_{n+1}(A)^{−1} if β_{n+1}(A) ≠ 0. For each A ∈ Fin_{W} (X), define k(A) ∈ ℕ by

A ∈ Fin^{k(A)}(k(A)B_{X}) \ Fin^{k(A)−1}((k(A) − 1)B_{X}).

Then, β_{n}(A) ≠ 0 for n < k(A) and β_{n}(A) = 0 for n ⩾ k(A),.

By Proposition 4.4, there exists a closed embedding h : Y → S_{X}. Take v_{n} ∈ S_{X}, n ∈ ℕ ∪ \{0\}, so that ∥v_{n} − v_{m}∥ > \frac{1}{2} for n ≠ m. We define a map g : Y → Fin_{W} (X) by

g(z) = f(z) ∪ {(β_{n}(f(z))^{−1} + 2)v_{n},

(β_{n}(f(z))^{−1} + 1)v_{n} + \frac{1}{8}h(z) | n < k(f(z))\}.

By (*), g is α-close to f. To see that g is injective, let z ≠ z′ ∈ Y. In case k(f(z)) ≠ k(f(z′)), assume k(f(z)) > k(f(z′)). Then,

(β_{k(f(z))−1}(f(z))^{−1} + 2)v_{k(f(z))−1} ∈ g(z) \ g(z′),

so g(z) ≠ g(z′). When k(f(z)) = k(f(z′)) = k, it follows that

\{(β_{k−1}(f(z))^{−1} + 2)v_{k−1}, (β_{k−1}(f(z))^{−1} + 1)v_{k−1} + \frac{1}{8}h(z)\}

≠ \{(β_{k−1}(f(z′))^{−1} + 2)v_{k−1}, (β_{k−1}(f(z′))^{−1} + 1)v_{k−1} + \frac{1}{8}h(z′)\},

which implies g(z) ≠ g(z′).

To see that g is a closed map, let y_{i} ∈ Y, i ∈ ℕ, and G ∈ Fin(X) such that g(y_{i}) → G. For each n ∈ ℕ ∪ {0}, let

b_{n} = lim inf_{i→∞} β_{n}(f(y_{i})) ∈ [0, (n + 1)^{−1}].

Then, 1 ⩾ b_{0} ⩾ b_{1} ⩾ ⋯ ⩾ 0. Moreover, b_{0} > 0. Otherwise, by taking a subsequence, we can assume that β(f(y_{i})) → 0. Then, it follows that f(y_{i}) → G, hence β(f(y_{i})) → β(G) ≠ 0, which is a contradiction.

Let b_{m} ≠ 0 and b_{m+1} = 0. Then, (b_{m}^{−1} + 2)v_{m} ∈ G ⊂ k(G)B_{X} because

d((b_{m}^{−1} + 2)v_{m}, G) = lim inf_{i→∞} d((b_{m}^{−1} + 2)v_{m}, g(y_{i})) = 0.

Hence, m + 3 ⩽ b_{m}^{−1} + 2 ⩽ k(G), that is, m ⩽ k(G) − 3. By taking a subsequence, we can assume that β_{m+1}(f(y_{i})) → 0. Since lim inf_{i∈ℕ} β(f(y_{i})) > 0, it follows that

d_{W} (f(y_{i}), Fin^{m+1}((m + 1)B_{X})) → 0.

Moreover, observe

lim inf_{i→ℕ} d_{W} (f(y_{i}), Fin^{m}(mB_{X})) ⩾ b_{m} > 0.

Then, we can choose

F_{i} ∈ Fin^{m+1}((m + 1)B_{X}) \ Fin^{m}(mB_{X}), i ∈ ℕ,

so that d_{W} (F_{i}, f(y_{i})) → 0. For each i ∈ ℕ, let

G_{i} = F_{i} ∪ {(β_{n}(f(y_{i}))^{−1} + 2)v_{n},

(β_{n}(f(y_{i}))^{−1} + 1)v_{n} + \frac{1}{8}h(y_{i}) | n < k(f(y_{i}))\}.

Then, G_{i} → G as i → ∞. Observe that

b_{m}^{−1} + 2 ⩾ ∥(β_{m}(f(y_{i}))^{−1} + 2)v_{n}∥ = β_{m}(f(y_{i}))^{−1} + 2
> β_{m}(f(y_{i}))^{−1} + \frac{7}{8}
⩾ ∥(β_{m}(f(y_{i}))^{−1} + 1)v_{m} + \frac{1}{8}h(y_{i})∥
⩾ β_{m}(f(y_{i}))^{−1} + \frac{7}{8} > m + \frac{3}{2}.

Since F_{i} ⊂ (m + 1)B_{X}, it follows that (h(y_{i}))_{i∈ℕ} is convergent. Hence, (y_{i})_{i∈ℕ} is convergent because h is a closed embedding.

8. The relative Wijsman topology

Let X = (X, d) be a separable metric space. For ℌ ⊂ Cld_{W} (X) and Y ⊂ X, we denote ℌ|Y = {A ∈ ℌ | A ⊂ Y }. Without any condition, we have Fin(X)|Y = Fin(Y) and Comp(X)|Y = Comp(Y) as sets. But, Cld(Y) = Cld(X)|Y if and only if Y is closed in X. As saw in Example, even if Y is closed in X, Fin_{W} (X)|Y ≠ Fin_{W} (Y) nor Cld_{W} (X)|Y ≠ Cld_{W} (Y) as spaces.

In this section, we give some sufficient conditions in order that Cld_{W} (X)|Y is an ANR or Fin_{W} (X)|Y is homotopy dense in Cld_{W} (X)|Y. Note that both Cld_{W} (X)|Y and Fin_{W} (X)|Y are Lawson semilattices of Cld_{W} (X).

Lemma 8.1. Let X be a separable metric space. For any path-connected closed subset Y ⊂ X, each A, B ∈ Cld_{W} (X)|Y can be connected by a path γ : I → Cld_{W} (X)|Y such that each γ(t) contains A or B.

Proof. It suffices to show that each A ∈ Cld(Y) can be connected to Y by a path γ : I → Cld_{W} (X) such that A ⊂ γ(t) ⊂ Y for every t ∈ I. Let {y_{n} | n ∈ ℕ} be dense in Y with y_{1} ∈ A. For each n ∈ ℕ, we have a path λ_{n} : [n, n + 1] → Y with λ_{n}(n) = y_{n} and λ(n + 1) = y_{n+1}. We define γ : I → Cld_{W} (X) by γ(0) = Y and

γ_{i}(t) = A ∪ ⋃_{n<m} λ_{n}([n, n + 1]) ∪ λ_{m}([m, t^{−1}]) for (m + 1)^{−1} < t ⩽ m^{−1}.

It is easy to verify the continuity of γ.

In the following, the case Y = X is Theorem IV for Cld_{W} (X).

Theorem 8.2. Let X be a separable metric space and Y a closed set of X such that for any finitely many open balls B_{1}, …, B_{n} in X, Y \ ⋃_{i=1}^{n} B_{i} has only finitely many path-components, all of which are closed in X. Then, the space Cld_{W} (X)|Y is an ANR.

Proof. By Corollary 5.4, it suffices to show that Cld_{W} (X)|Y is locally pathconnected. Let A ∈ Cld_{W} (X) and U a neighborhood of A in Cld_{W} (X). Then,

A ∈ ⋂_{i<k} U^{−}(p_{i}, r_{i}) ∩ ⋂_{j<l} U^{+}(q_{j}, s_{j}) ⊂ U,

for some p_{i}, q_{j} ∈ X and r_{i}, s_{j} > 0. Choose ε > 0 so that A ∈ U^{+}(q_{j}, s_{j} + ε) for every j < l.

Let Y_{0}, …, Y_{m−1} be all path-components of Y \ ⋃_{j<l} B(q_{j}, s_{j} + ε), where Y_{i} ∩ A ≠ ∅ for i < m_{0} and Y_{j} ∩ A = ∅ for j ⩾ m_{0}. Observe that each Y_{i} is open in Y \ ⋃_{j<l} B(q_{j}, s_{j} + ε) and A is strongly disjoint from ⋃_{j<l} B(q_{j}, s_{j} + ε). For each i < m_{0}, we can choose p_{k+i} ∈ Y_{i} ∩ A and r_{k+i} > 0 so that B(p_{k+i}, r_{k+i}) ∩ Y ⊂ Y_{i}. Since A ⊂ ⋃_{i<m_{0}} Y_{i}, it follows that ⋃_{j⩾m_{0}} Y_{i} is covered by open balls which are strongly disjoint from A. It can be assumed that m_{0} ⩽ j < m_{1} if and only if Y_{j} cannot be covered by finite open balls which are strongly disjoint from A. We can find t_{0}, …, t_{v−1} ∈ X and δ_{0}, …, δ_{v−1} > 0 such that

⋃_{m_{1}⩽j<m} Y_{j} ⊂ ⋃_{i<v} B(t_{i}, δ_{i}) and A ∈ U^{+}(t_{i}, δ_{i}) for each i < v.^{6}

Thus, we have the neighborhood V of A defined by

V = ⋂_{i<k+m_{0}} U^{−}(p_{i}, r_{i}) ∩ ⋂_{j<l} U^{+}(q_{j}, s_{j} + ε) ∩ ⋂_{i<v} U^{+}(t_{i}, δ_{i}) ⊂ U.

It suffices to show that each B ∈ V|Y can be connected to A by a path in U|Y. Then, B ⊂ ⋃_{j<m_{1}} Y_{j} and B ∩ Y_{i} ⊃ B ∩ B(p_{k+i}, r_{k+i}) ≠ ∅ for every i < m_{0}. By Lemma 8.1, there are paths γ_{i} : I → Cld_{W} (Y_{i}), i < m_{0}, such that γ_{i}(0) = A ∩ Y_{i}, γ_{i}(1) = B ∩ Y_{i} and each γ_{i}(t) contains A ∩ Y_{i} or B ∩ Y_{i}. Let S = {j ⩾ m_{0} | B ∩ Y_{j} ≠ ∅}. For each j ∈ S, take z_{j} ∈ B ∩ Y_{j} and define B^{∗} = {z_{j} | j ∈ S}. By Lemma 8.1, we have paths φ_{j} : I → Cld_{W} (Y_{j}), j ∈ S, such that φ_{j}(0) = z_{j} and φ(1) = B ∩ Y_{j}. Then, we have a path γ : I → U|Y from A ∪ B^{∗} to B defined by

6See footnote 3.

γ(t) = ⋃_{i<m_{0}} γ_{i}(t) ∪ ⋃_{j∈S} φ_{j}(t).

For each j ∈ S, we will construct an infinite path ψ_{j} : [1, ∞) → Y_{j} with ψ_{j}(1) = z_{j} and the following property:

(*) for each n ∈ ℕ, there is some t_{0} > 0 such that

d(x_{i}, ψ_{j}(t)) > d(x_{i}, A) − 1/n for i ⩽ n and t ⩾ t_{0},

where {x_{i} | i ∈ ℕ} is a countable dense set in X to define the metric d_{W}. Then, we can define a path ψ : I → V from A to A∪B^{∗} as follows: ψ(0) = A and

ψ(t) = A ∪ ⋃_{j∈S} ψ_{j}(t^{−1}) for t > 0.

The continuity of λ at 0 can be verified as follows: For each n ∈ ℕ, we use (*) to find t_{0} > 0 so that

d(x_{i}, ψ_{j}(t)) > d(x_{i}, A) − 1/n for i ⩽ n, t ⩾ t_{0} and j ∈ S.

Then, t < t_{0}^{−1} implies |d(x_{i}, ψ_{j}(t)) − d(x_{i}, A)| < 1/n for i ⩽ n, hence d_{W} (ψ(t), A) < 1/n.

Now, for j ∈ S, we construct an infinite path ψ_{j} : [1, ∞) → Y_{j}. Enumerate as B_{1}, B_{2}, … all open balls of the form B(x_{i}, α), where 0 < α < d(x_{i}, A) and α ∈ ℚ. By the assumption, for each n ∈ ℕ, Y_{j} \ (B_{1} ∪ ⋯ ∪ B_{n}) has finitely many path-components H_{0}^{n}, …, H_{a(n)−1}^{n}. Let T = {H_{i}^{n} | n ∈ ℕ, i < a(n)}. Since every B_{i} is strongly disjoint from A, Y_{j} cannot be covered by finitely many B_{i}’s, hence T is infinite. Thus, (T, ⊃) is a finitely-branching infinite tree (i.e., each element of T has only finitely many immediate successors). By König’s lemma, T contains an infinite branch Y_{j} ⊃ H_{i(1)}^{1} ⊃ H_{i(2)}^{2} ⊃ · · ·. For each n ∈ ℕ, pick y_{n} ∈ H_{i(n)}^{n} and a path f_{n} : [n − 1, n] → H_{i(n−1)}^{n−1} such that f_{n}(n − 1) = y_{n−1} and f_{n}(n) = y_{n}, where H_{i(0)}^{0} = Y_{j} and y_{0} = z_{j}. By joining all paths f_{1}, f_{2}, …, we can obtain a path ψ_{j} : [1, +∞) → Y_{j} with f_{y}(1) = z_{j}.

For each i ⩽ n, choose α_{i} ∈ ℚ so that d(x_{i}, A) − 1/n < α_{i} < d(x_{i}, A). Then we have m ∈ ℕ such that all balls B(x_{1}, α_{1}), …, B(x_{n}, α_{n}) appear in B_{0}, …, B_{m−1}. For t < 1/m, we have l ⩾ m such that l < t^{−1} ⩽ l + 1, whence

ψ_{j}(t) = f_{l+1}(t^{−1}) ∈ H_{i(l)}^{n} ⊂ Y_{j} \ (B(x_{1}, α_{1}) ∪ ⋯ ∪ B(x_{n}, α_{n})).

Therefore, d(x_{i}, ψ_{j}(t)) ⩾ α_{i} > d(x_{i}, A) − 1/n for every i ⩽ n. Thus ψ satisfies (*).

In the above, we use Lemma 6.2 instead of Lemma 8.1 to obtain the following, in which the case Y = X is Theorem IV for Fin_{W} (X):

Theorem 8.3. Let X be a separable metric space and Y a path-connected subset of X such that for any finitely many open balls B_{1}, …, B_{n} in X, Y \ ⋃_{i=1}^{n} B_{i} has only finitely many path-components which are open in Y \ ⋃_{i=1}^{n} B_{i}. Then, the space Fin_{W} (X)|Y is an ANR.

In the same setting as Theorem 8.2, Theorem 6.3 (or Theorem III) can be generalized as follows:

Theorem 8.4. Let X be a separable metric space and Y a path-connected subset of X such that for any finitely many open balls B_{1}, …, B_{n} in X, Y \ ⋃_{i=1}^{n} B_{i} has finitely many path-components which are compact or unbounded (resp. all are unbounded). Then, Fin_{W} (X)|Y is homotopy dense in Cld_{W} (X)|Y and Cld_{W} (X)|Y is an ANR (resp. an AR).

The proof of this theorem is left to the readers.

References

[1] T. Banakh, Characterization of spaces admitting a homotopy dense embedding into a Hilbert manifold, Topology Appl. 86 (1998), 123–131.

[2] T. Banakh, M. Kurihara and K. Sakai, Hyperspaces of normed linear spaces with the Attouch-Wets topology, submitted.

[3] T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, VNTL Publishers, Lviv, 1996.

[4] G. Beer, Topologies on Closed and Closed Convex Sets, MIA 268, Kluwer Acad. Publ., Dordrecht, 1993.

[5] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan Math. J. 33 (1986), 291–313.

[6] D. Curtis, T. Dobrowolski and J. Mogilski, Some applications of the topological characterizations of the sigma-compact spaces ℓ_{f}^{2} and Σ, Trans. Amer. Math. Soc. 284 (1984), 837–846.

[7] Doug Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to ℵ_{0}-dimensional linear metric spaces, Topology Appl. 19 (1985), 251–260.

[8] D.W. Curtis and R.M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19–38.

[9] S.-T. Hu, Theory of Retracts, Wayne State Univ. Press, Detroit, 1965.

[10] A.S. Kechris, Classical Descriptive Set Theory, GTM 156, Springer-Verlag New York, Inc., New York, 1995.

[11] W. Kubiś, Completely metrizable absolute neighborhood retracts, Preprint.

[12] J.D. Lawson, Topological semilattices with small subsemilattices, J. London Math. Soc. (2) 1 (1969), 719–724.

[13] R. Lowen and M. Sioen, The Wijsman and Attouch-Wets topologies on hyperspaces revisited, Topology Appl. 70 (1996), 179–197.

[14] J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library 43, Elsevier Sci. Publ. B.V., Amsterdam, 1989.

[15] K. Sakai, The completions of metric ANR’s and homotopy dense subsets, J. Math. Soc. Japan 52 (2000), 835–846.

[16] K. Sakai and M. Yaguchi, Hyperspaces of infinite-dimensional Banach spaces with the Attouch-Wets topology, Preprint.

[17] K. Sakai and Z. Yang, Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, to appear in Topology Appl.

[18] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of ℓ_{2}-manifolds, Fund. Math. 101 (1978), 93–110.

[19] H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247– 262;

[20] H. Toruńczyk, A correction of two papers concerning Hilbert manifolds, Fund. Math. 125 (1985), 89–93.

[21] M. Wojdysławski Rétractes absoulus et hyperespaces des continus, Fund. Math. 32 (1939), 184–192.

(W. Kubiś) Institute of Mathematics, University of Silesia, Katowice, Poland and Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel

   E-mail address: kubis@math.bgu.ac.il

(K. Sakai) Institute of Mathematics, University of Tsukuba, Tuskuba, 305-8571, Japan

   E-mail address: sakaiktr@sakura.cc.tsukuba.ac.jp

(M. Yaguchi) Institute of Mathematics, University of Tsukuba, Tuskuba, 305-8571, Japan

   E-mail address: masato@math.tsukuba.ac.jp