Spaces of upper semicontinuous multi-valued functions on complete metric spaces

160 (1999)

Spaces of upper semicontinuous

multi-valued functions on complete metric spaces

by

Katsuro S a k a i and Shigenori U e h a r a (Tsukuba)

Abstract. Let X = (X, d) be a metric space and let the product space X × R be en- dowed with the metric %((x, t), (x

, t

)) = max{d(x, x

), |t − t

|}. We denote by USCC

(X) the space of bounded upper semicontinuous multi-valued functions ϕ : X → R such that each ϕ(x) is a closed interval. We identify ϕ ∈ USCC

(X) with its graph which is a closed subset of X × R. The space USCC

(X) admits the Hausdorff metric induced by %. It is proved that if X = (X, d) is uniformly locally connected, non-compact and complete, then USCC

(X) is homeomorphic to a non-separable Hilbert space. In case X is separable, it is homeomorphic to `

(2

).

1. Introduction. Let X = (X, d) be a metric space and let the product space X × R be endowed with the metric

%((x, t), (x ⁰ , t ⁰ )) = max{d(x, x ⁰ ), |t − t ⁰ |}.

A multi-valued function ϕ : X → R is said to be bounded if the image ϕ(X) = S

x∈X ϕ(x) is bounded. For any multi-valued function ϕ : X → R such that each ϕ(x) is compact, ϕ is upper semicontinuous (u.s.c.) if and only if the graph of ϕ is closed in X ×R. Such a ϕ can be regarded as a closed set in X × R. We denote by USCC _B (X) the space of bounded u.s.c. multi- valued functions ϕ : X → R such that each ϕ(x) is non-empty, compact and connected, that is, a closed interval. The topology for USCC B (X) is induced by the Hausdorff metric

% _H (ϕ, ψ) = max{sup

z∈ϕ %(z, ψ), sup

z∈ψ

%(z, ϕ)},

where %(z, ψ) = inf _z

_∈ψ %(z, z ⁰ ). Since ϕ and ψ are bounded, % _H (ϕ, ψ) < ∞ can be defined. In case X is compact, every u.s.c. multi-valued function

1991 Mathematics Subject Classification: 54C60, 57N20, 58C06, 58D17.

Key words and phrases: space of upper semicontinuous multi-valued functions, hy- perspace of non-empty closed sets, Hausdorff metric, Hilbert space, uniformly locally connected.

[199]

ϕ : X → R is bounded, so we write USCC B (X) = USCC(X). Let USCC(X, I) = {ϕ ∈ USCC _B (X) | ϕ(X) ⊂ I},

where I = [0, 1]. In case X is non-compact, as will be seen, the topology for USCC _B (X) (or USCC(X, I)) depends on the metric d.

Fedorchuk [Fe _1,2 ] proved that if X is infinite, locally connected and compact then USCC(X, I) is homeomorphic to (≈) the Hilbert cube Q = [−1, 1] ^ω and USCC(X) ≈ Q \ {0} (≈ Q × [0, 1)) (cf. [SU, Appendix]). In this paper, we consider the case where X is non-compact but complete. We say that X is uniformly (or d-uniformly) locally connected if, for each ε > 0, there is δ > 0 such that each pair of points x, x ⁰ ∈ X with d(x, x ⁰ ) < δ are contained in some connected set in X with diameter < ε. Let m (or ` <sub>∞</sub> ) be the Banach space of bounded sequences in R with the sup-norm. Note that m is non-separable. Indeed, m ≈ ` ₂ (2 ^N ) [BP, Ch. VII, Theorem 6.1].

By applying Toru´ nczyk’s characterization of Hilbert spaces [To ₃ ] (cf. [To ₄ ]), we prove the following:

Main Theorem. If X = (X, d) is a uniformly locally connected, non- compact and complete metric space, then USCC(X, I) and USCC _B (X) are homeomorphic to a non-separable Hilbert space. In case X is separable,

USCC(X, I) ≈ USCC _B (X) ≈ m ≈ ` ₂ (2 ^N ).

In the above, the word “uniformly” cannot be removed, that is, the Main Theorem is not valid for a locally connected complete metric space X with no isolated points.

Example. The following closed subspace X of Euclidean plane R ² is locally path-connected and has no isolated points, but USCC(X, I) and USCC _B (X) are not locally connected, hence they are not ANR’s:

X = R × {0} ∪ [

n∈N

{n, n + 2 ⁻ⁿ } × I ⊂ R ² .

P r o o f. We define a map f : X → I by f (s, t) =

( 2t if s ∈ N and 0 ≤ t ≤ 1/2, 1 if s ∈ N and 1/2 ≤ t ≤ 1, 0 otherwise.

For each ε > 0, choose n 0 ∈ N so that 2 ⁻ⁿ

< ε, and define g : X → I by

g(s, t) =

 



 

0 if s = n ₀ ,

2t if s = n ₀ + 2 ⁻ⁿ

and 0 ≤ t ≤ 1/2, 1 if s = n 0 + 2 ⁻ⁿ

and 1/2 ≤ t ≤ 1, f (s, t) otherwise.

Then % H (f, g) = 2 ⁻ⁿ

< ε but g cannot be connected with f by any path in

USCC _B (X) with diameter < 1/2.

In the above, X ≈ Y = R × {0} ∪ N × I ⊂ R ² , but USCC(X, I) 6≈

USCC(Y, I) because USCC(Y, I) ≈ ` ₂ (2 ^N ) by the Main Theorem.

Throughout the paper, the open ε-ball in X = (X, d) centered at x ∈ X is denoted by B(x, ε) (or B d (X, ε)) and the closure of B(x, ε) in X by B(x, ε). On the other hand, to avoid confusion, the ε-neighborhood of a subset F ⊂ X in X is denoted by N (F, ε) (or N _d (F, ε)), that is,

N (F, ε) = [

x∈F

B(x, ε) = {y ∈ X | d(y, F ) < ε} ⊂ X.

For F ⊂ X × R and A ⊂ X, we define F |A = F ∩ pr ⁻¹ _X (A) = F ∩ A × R and F (A) = pr _R (F |A), where pr _X : X × R → X and pr _R : X × R → R are the projections. In case A = {x}, we write F |{x} = F |x and F ({x}) = F (x).

1. Relations among C _B (X), USCC _B (X) and 2 ^X×R . For a metric space X = (X, d), let (2 ^X ) m denote the hyperspace of non-empty bounded closed subsets of X with the Hausdorff metric d _H defined by d (cf. [Ku, p. 214]).

If X is complete, then so is (2 ^X ) _m [Ku, p. 407]. In case X is compact, (2 ^X ) m is the hyperspace exp(X) of non-empty compact subsets of X. Let 2 ^X be the totality of non-empty closed subsets of X. When X is unbounded, 2 ^X 6= (2 ^X ) _m and d _H is not a metric on the whole 2 ^X (e.g., X 6∈ (2 ^X ) _m and d H ({x}, X) = ∞ for any x ∈ X), but d H induces the topology on 2 ^X . In fact, A ∈ 2 ^X has a neighborhood base consisting of

{B ∈ 2 ^X | d H (A, B) < ε} (= {B ∈ 2 ^X | A ⊂ N d (B, ε), B ⊂ N d (A, ε)}).

The spaces USCC(X, I) ⊂ USCC _B (X) are regarded as subspaces of the hyperspace 2 ^X×R . Note that USCC(X, I) 6⊂ (2 ^X×R ) m if X is unbounded, and that % _H is not a metric on 2 ^X×R but it is a metric on USCC _B (X).

One should remark that a different metric d ⁰ on X defines not only a different space (2 ^X ) m but also a different topology on 2 ^X even if d ⁰ induces the same topology of X as d. However, if d ⁰ is uniformly equivalent to d, then d ⁰ _H induces the same topology on 2 ^X as d H . Let d ^∗ be the bounded metric on X defined by d ^∗ (x, y) = min{1, d(x, y)}. Note that every closed subset on X is bounded with respect to d ^∗ . Since d ^∗ _H is a metric on the whole 2 ^X , the space 2 ^X is metrizable. Moreover, if d is complete, then so is d ^∗ , hence d ^∗ _H is also complete (cf. [Ku, p. 407]).

The following is elementary, but we give a proof for completeness.

1.1. Lemma. Let ϕ ∈ USCC _B (X) and A ⊂ X. If A is connected, then so is the image ϕ(A).

P r o o f. Assume that ϕ(A) is disconnected. Then there is t ∈ R \ ϕ(A)

such that (−∞, t)∩ϕ(A) 6= ∅ and (t, ∞)∩ϕ(A) 6= ∅, whence ϕ(x) ⊂ (−∞, t)

or ϕ(x) ⊂ (t, ∞) for each x ∈ A because of the connectedness of ϕ(x). Let

U = {x ∈ X | ϕ(x) ⊂ (−∞, t)} and V = {x ∈ X | ϕ(x) ⊂ (t, ∞)}. Then

U ∩ V = ∅, A ⊂ U ∩ V , A ∩ U 6= ∅ and A ∩ V 6= ∅. Since ϕ is u.s.c., these U and V are open sets in X. This contradicts the connectedness of A. Hence ϕ(A) is connected.

Without any completeness condition, the following can be proved (cf.

[FK, Theorem 3.3(a)]).

1.2. Proposition. If X is locally connected, then USCC _B (X) is closed in 2 ^X×R , hence USCC(X, I) is closed in 2 ^X×I .

P r o o f. Let ϕ ∈ cl ₂

USCC B (X). Then, as is easily observed, ϕ ⊂ X × [−a, a] for some a > 0. If ϕ(x) = ∅ (i.e., ϕ ∩ {x} × R = ∅), then B(x, ε) × R ∩ ϕ = ∅ for some ε > 0. For any ψ ∈ USCC _B (X), since ψ(x) 6= ∅, we have % H (ψ, ϕ) ≥ ε, which is a contradiction. Therefore, ϕ(x) 6= ∅ for every x ∈ X. Since ϕ is closed in X × R, it follows that ϕ : X → R is u.s.c. We show that each ϕ(x) is connected, which implies that ϕ ∈ USCC _B (X).

Assume that some ϕ(x 0 ) is not connected. Then we can find some t 1 <

t ₀ < t ₂ such that t ₁ , t ₂ ∈ ϕ(x ₀ ) and t ₀ 6∈ ϕ(x ₀ ). Choose ε > 0 so that B(x ₀ , 2ε) × (t ₀ − ε, t ₀ + ε) ∩ ϕ = ∅,

whence %((x, t 0 ), ϕ) ≥ ε for each x ∈ B(x 0 , ε). Since X is locally connected, x ₀ has a connected neighborhood U ⊂ B(x ₀ , ε). Then U contains some B(x ₀ , δ) ⊂ U , whence δ ≤ ε. For each ψ ∈ USCC _B (X) with % _H (ψ, ϕ) < δ, we have some (x i , s i ) ∈ ψ, i = 1, 2, such that d(x i , x 0 ) < δ and |t i − s i | <

δ, whence x ₁ , x ₂ ∈ U , s ₁ < t ₀ and s ₂ > t ₀ . Since ψ(U ) is connected by Lemma 1.1, it follows that t 0 ∈ [s 1 , s 2 ] ⊂ ψ(U ), that is, t 0 = ψ(x) for some x ∈ U ⊂ B(x ₀ , ε). Then % _H (ψ, ϕ) ≥ %((x, t ₀ ), ϕ) ≥ ε, which is a contradiction. Therefore, every ϕ(x) is connected. Thus ϕ ∈ USCC _B (X).

By the remark at the beginning of this section, the statement below easily follows from Proposition 1.2.

1.3. Corollary. If X is complete and locally connected, then USCC _B (X) is complete, hence so is USCC(X, I).

Let C _B (X) be the Banach space of bounded continuous real-valued func- tions on X with the sup-norm ( ¹ ). Let C(X, I) = {f ∈ C B (X) | f (X) ⊂ I}.

In case X is compact, every continuous real-valued function on X is bounded, and therefore we write C B (X) = C(X). For a compact space X, Fedorchuk [Fe _1,2 ] proved that if X is locally connected and has no isolated points then C(X) and C(X, I) are dense in USCC(X) and USCC(X, I), respectively.

This was generalized in [FK] to non-compact spaces with some complete- ness condition. Here we give a proof without local connectedness or any completeness condition.

(

) As in [FK, Remark 3.6], although C

(X) ⊂ USCC

(X), the Banach space C

(X)

is not a subspace of USCC

(X) in case X is non-compact.

1.4. Lemma. For each ϕ ∈ USCC(X, I) and ε > 0, there exists a lower semicontinuous (l.s.c.) multi-valued function ϕ _ε : X → I such that each ϕ _ε (x) is a closed interval, ϕ ⊂ ϕ _ε and % _H (ϕ, cl _X×I ϕ _ε ) ≤ ε.

P r o o f. For each x ∈ X, let

V _x = (min ϕ(x) − ε, max ϕ(x) + ε) ∩ I.

Since ϕ is u.s.c., we can choose δ _x > 0 so that δ _x ≤ ε and ϕ(x ⁰ ) ⊂ V _x if x ⁰ ∈ B(x, δ x ) (i.e., d(x, x ⁰ ) < δ x ). Let ψ : X → I be the multi-valued function defined by

ψ(x) = [

{V _y | d(x, y) < δ _y } for each x ∈ X.

We define the multi-valued function ϕ _ε : X → I by ϕ _ε (x) = cl _I ψ(x). Then ϕ ⊂ ψ ⊂ ϕ _ε . As is easily observed, % _H (ϕ, cl _X×I ψ) ≤ ε. Since cl _X×I ϕ _ε = cl X×I ψ, we have % H (ϕ, cl X×I ϕ ε ) ≤ ε. If d(x, y) < δ y then ϕ(x) ⊂ V y . Since ϕ(x) and V _y are connected, each ψ(x) is connected, hence so is ϕ _ε (x).

To see that ϕ _ε is l.s.c., let V be an open set in I and x ∈ X such that ϕ _ε (x) ∩ V 6= ∅. Then we have t ∈ ψ(x) ∩ V . By the definition of ψ, we can find y ∈ X such that d(x, y) < δ y and t ∈ V y . If d(x, x ⁰ ) < δ y − d(x, y) then d(x ⁰ , y) < δ _y , hence V _y ⊂ ψ(x ⁰ ) ⊂ ϕ _ε (x ⁰ ) by the definition. Thus we have t ∈ ϕ _ε (x ⁰ ) ∩ V . Therefore, ϕ _ε : X → I is l.s.c.

Remark. In the above, ϕ ε 6= cl X×I ψ. For example, let ϕ = I × {0} ∪ [1/2, 1] × I ∈ USCC(I, I) and ε = 1/2. Then V _x = [0, 1/2) for x < 1/2 and V _x = I for x ≥ 1/2. Define ψ as above by using

δ _x =

 1/2 − x if x < 1/2, 1/2 if x ≥ 1/2.

Observe that d(0, y) < δ _y implies y < 1/2, and that d(x, 1/2) < δ _1/2 = 1/2 for x 6= 0, 1. Therefore, ψ = {0} × [0, 1/2) ∪ (0, 1] × I = I ² \ {0} × [1/2, 1], hence cl _X×I ψ = I ² . On the other hand, ϕ _1/2 = {0} × [0, 1/2] ∪ (0, 1] × I because ϕ _1/2 (x) = cl _I ψ(x) for each x ∈ I.

1.5. Theorem. The following conditions are equivalent for any metric space X = (X, d):

(a) C(X, I) is dense in USCC(X, I);

(b) C _B (X) is dense in USCC _B (X);

(c) X has no isolated points.

P r o o f. (a)⇒(b). This follows from the fact that each ϕ ∈ USCC _B (X) is contained in some USCC B (X, [−a, a]).

(b)⇒(c). When X has an isolated point x 0 , let ϕ = X × {0} ∪ {x 0 } × I ∈

USCC _B (X). Then, as is easily observed,

% H (ϕ, f ) ≥ min{1/2, d(x 0 , X \ {x 0 })} > 0 for any f ∈ C B (X), which implies that C B (X) is not dense in USCC B (X).

(c)⇒(a). For each ϕ ∈ USCC(X, I) and ε > 0, let ϕ ε : X → I be the l.s.c. multi-valued function obtained by Lemma 1.4. Choose a discrete closed subset D of ϕ so that %((x, t), D) < ε/2 for any (x, t) ∈ ϕ, whence

% H (ϕ, D) < ε/2. Note that pr _X |D is finite-to-one and pr _X (D) is discrete in X. Since ϕ _ε is l.s.c. and X has no isolated points, for each (x, t) ∈ ϕ _ε there are infinitely many y ∈ X such that

d(x, y) < ε/2 and ϕ _ε (y) ∩ (t − ε/2, t + ε/2) 6= ∅.

Then we can construct a discrete closed subset f of ϕ _ε such that pr _X |f is injective and % H (D, f ) < ε/2, hence % H (ϕ, f ) < ε. Then A = pr _X (f ) is discrete in X and f : A → I is a map ( ² ) which is a selection for ϕ _ε |A (i.e., f (x) ∈ ϕ _ε (x) for each x ∈ A). By Michael’s Selection Theorem [Mi], we can extend f to e f ∈ C(X, I) which is a selection for ϕ ε . For any (x, t) ∈ ϕ, we have %((x, t), e f ) ≤ %((x, t), f ) ≤ % H (ϕ, f ) < ε. Since e f ⊂ cl X×I ϕ ε and

% H (ϕ, cl X×I ϕ ε ) ≤ ε, it follows that %((x, t), ϕ) ≤ ε for any (x, t) ∈ e f . Thus

% _H ( e f , ϕ) ≤ ε. Consequently, ϕ ∈ cl ₂

C(X, I).

Combining Theorem 1.5 with Proposition 1.2, we have the following corollary:

1.6. Corollary. For any locally connected metric space X with no iso- lated points, USCC B (X) (resp. USCC(X, I)) is the closure of C B (X) (resp.

C(X, I)) in 2 ^X×R (resp. 2 ^X×I ).

One should notice that no completeness is assumed above (cf. [FK, The- orem 3.3(a)]).

2. The AR-property of USCC _B (X) and USCC(X, I). In this section, using Borges’ characterization of AR’s in [Bo], we prove that USCC B (X) and USCC(X, I) are AR’s if X = (X, d) is uniformly locally connected.

Now, we define a new metric d _c on X as follows:

d _c (x, x ⁰ ) =

 inf{diam _d C | C ∈ C(x, x ⁰ )} if C(x, x ⁰ ) 6= ∅,

1 otherwise,

where

C(x, x ⁰ ) = {C ⊂ X | C is connected, x, x ⁰ ∈ C and diam C < 1}.

As is easily observed, if X is uniformly locally connected, then d _c is uniformly equivalent to d, hence d c induces the same topology on 2 ^X×R as d. Then, by replacing d with d _c , we can assume that

(

) Recall that a map is identified with its graph.

(∗) each pair of points x, x ⁰ ∈ X with d(x, x ⁰ ) < ε < 1 are contained in a connected set C in X with diam C < ε.

2.1. Lemma. Condition (∗) implies the following condition:

(]) N _% (ϕ, ε)(x) is connected for each ϕ ∈ USCC _B (X), 0 < ε < 1 and x ∈ X.

P r o o f. Let t 1 , t 2 ∈ N % (ϕ, ε)(x) and t 1 < t < t 2 . Then there are x 1 , x 2 ∈ X and s _i ∈ ϕ(x _i ) (i = 1, 2) such that d(x _i , x) < ε and |s _i − t _i | < ε. Let

s = t ₂ − t

t ₂ − t ₁ s ₁ + t − t ₁ t ₂ − t ₁ s ₂ .

By (∗), X has connected subsets C ₁ and C ₂ such that x _i , x ∈ C _i and diam C _i

< ε. Since C = C 1 ∪ C 2 is connected, s ∈ ϕ(x 0 ) for some x 0 ∈ C by Lemma 1.1. Then d(x ₀ , x) < ε. Observe that

t = t ₂ − t

t ₂ − t ₁ t ₁ + t − t ₁ t ₂ − t ₁ t ₂ . It then follows that

|s − t| ≤ t 2 − t

t ₂ − t ₁ |s ₁ − t ₁ | + t − t 1

t ₂ − t ₁ |s ₂ − t ₂ | < ε.

So (x, t) ∈ N % (ϕ, ε), i.e., t ∈ N % (ϕ, ε)(x). Thus, N % (ϕ, ε)(x) is connected.

We denote by ∆ ⁿ⁻¹ the standard (n − 1)-simplex in R ⁿ , that is,

∆ ⁿ⁻¹ = n

(t 1 , . . . , t n ) ∈ R ⁿ  

 t i ≥ 0, X n i=1

t i = 1 o

.

A space Y is called hyper-connected if there are functions h n : Y ⁿ ×∆ ⁿ⁻¹ → Y (n ∈ N) which satisfy the following conditions:

(i) if t i = 0 then

h n (y 1 , . . . , y n ; t 1 , . . . , t n )

= h _n−1 (y ₁ , . . . , y _i−1 , y _i+1 , . . . , y _n ; t ₁ , . . . , t _i−1 , t _i+1 , . . . , t _n );

(ii) ∆ ⁿ⁻¹ 3 (t ₁ , . . . , t _n ) 7→ h _n (y ₁ , . . . , y _n ; t ₁ , . . . , t _n ) ∈ Y is continuous for each (y 1 , . . . , y n ) ∈ Y ⁿ ;

(iii) each neighborhood U of y ∈ Y contains a neighborhood V of y such that h _n (V ⁿ × ∆ ⁿ⁻¹ ) ⊂ U for every n ∈ N.

Notice that h n need not be continuous. It was proved by C. R. Borges [Bo]

that a metrizable space X is an AR if and only if X is hyper-connected ( ³ ).

We apply this characterization to prove the following:

(

) R. Cauty [Ca] introduced a local hyper-connectedness different from the one of

[Bo] and showed that a metrizable space X is an ANR if and only if X is locally hyper-

connected. The results of [Bo] and [Ca] hold for stratifiable spaces.

2.2. Theorem. For any uniformly locally connected metric space X = (X, d), USCC _B (X) and USCC(X, I) are AR’s.

P r o o f. Since USCC(X, I) is a retract of USCC _B (X), it suffices to show that USCC _B (X) is an AR.

By replacing the metric d with d _c , we can assume condition (∗). Each point of ∆ ⁿ⁻¹ \ {b n−1 } can be uniquely represented as follows:

(1 − t)b _n−1 + z, z ∈ ∂∆ ⁿ⁻¹ , 0 < t ≤ 1,

where b n−1 is the barycenter of ∆ ⁿ⁻¹ and ∂∆ ⁿ⁻¹ is the boundary of ∆ ⁿ⁻¹ . We inductively define

h n : USCC B (X) ⁿ × ∆ ⁿ⁻¹ → USCC B (X) (n ∈ N).

First, let h ₁ (ϕ, 1) = ϕ for every ϕ ∈ USCC _B (X). Assume that h ₁ , . . . , h _n−1 have been defined, and define h n as follows:

h _n (ϕ ₁ , . . . , ϕ _n ; b _n−1 )(x) = h

min [ n i=1

ϕ _i (x), max [ n i=1

ϕ _i (x) i

and, for z ∈ ∂∆ ⁿ⁻¹ and 0 < t ≤ 1, h _n (ϕ ₁ , . . . , ϕ _n ; (1 − t)b _n−1 + tz)(x)

= (1 − t)h n (ϕ 1 , . . . , ϕ n ; b n−1 )(x) + th n (ϕ 1 , . . . , ϕ n ; z)(x), where h _n (ϕ ₁ , . . . , ϕ _n ; z) is defined by condition (i). Then conditions (i) and (ii) are clearly satisfied. We show that

h _n (B _%

(ϕ, ε) ⁿ × ∆ ⁿ⁻¹ ) ⊂ B _%

(ϕ, ε)

for each ϕ ∈ USCC _B (X) and 0 < ε < 1. For ϕ ₁ , . . . , ϕ _n ∈ B _%

(ϕ, ε) and z ∈ ∆ ⁿ⁻¹ , since ϕ ₁ , . . . , ϕ _n ⊂ N _% (ϕ, ε), it follows from Lemma 2.1 and the definition of h n that

h _n (ϕ ₁ , . . . , ϕ _n ; z) ⊂ h _n (ϕ ₁ , . . . , ϕ _n ; b _n−1 ) ⊂ N _% (ϕ, ε).

On the other hand, since h n (ϕ 1 , . . . , ϕ n ; z) contains some ϕ i and since ϕ ⊂ N _% (ϕ _i , ε), we have ϕ ⊂ N _% (h _n (ϕ ₁ , . . . , ϕ _n ; z), ε). Therefore,

% H (h n (ϕ 1 , . . . , ϕ n ; z), ϕ) < ε (i.e., h n (ϕ 1 , . . . , ϕ n ; z) ∈ B %

(ϕ, ε)).

Thus (iii) also holds. Consequently, USCC _B (X) is hyper-connected, hence it is an AR.

3. Proof of Main Theorem. We use the following variant of Toru´ n- czyk’s characterization of Hilbert space [To ₃ ] (cf. [To ₄ ]):

3.1. Lemma. Let A be a discrete space and H = (H, d) a complete AR

with weight w(H) = card A. Then H ≈ ` 2 (A) if and only if the following

condition is satisfied:

(∗∗) for any open cover U of H, there exists a map f : H × A → H such that {f _a (H) | a ∈ A} is discrete in H and each f _a is U-close to id, where f _a : H → H is defined by f _a (x) = f (x, a).

P r o o f. Obviously, (∗∗) implies conditions (∗1) and (∗2) in [To 3 , Theo- rem 3.1] (cf. [To ₄ ]), hence we have the “if” part. The “only if” part easily follows from the fact that the projection pr ₁ : H × H → H onto the first factor is a near homeomorphism (cf. [Sc]).

3.2. Lemma. Assume condition (∗) of §2 is satisfied, X has no isolated points, and there exist D ⊂ X and δ, ε ∈ (0, 1) such that d(a, a ⁰ ) ≥ ε for a 6= a ⁰ ∈ D and each a ∈ D has a connected neighborhood with di- ameter > δ. Then, for any open cover U of USCC B (X), there exists a map h : USCC _B (X) × 2 ^D → USCC _B (X) such that {h _F (USCC _B (X)) | F ∈ 2 ^D } is discrete in USCC _B (X) and each h _F is U-close to id, where h F : USCC B (X) → USCC B (X) is defined by h F (ϕ) = h(ϕ, F ).

P r o o f. Let V be an open star-refinement of U. Since USCC _B (X) is an AR (Theorem 2.2), we have a simplicial complex K with maps

p : USCC _B (X) → |K| and q : |K| → USCC _B (X)

such that qp is V-close to id. Let α : USCC _B (X) → (0, 1) be a map such that α(ϕ) < min{δ, ε} for each ϕ ∈ USCC B (X) and

{B _%

(ϕ, 2α(ϕ)) | ϕ ∈ USCC _B (X)} ≺ V.

By subdividing K, we can assume the following two conditions:

(1) diam %

q(σ) < ¹ ₈ αq(y) if y ∈ σ ∈ K;

(2) αq(y) < 2αq(y ⁰ ) if y, y ⁰ ∈ σ ∈ K.

In fact, for each ϕ ∈ USCC _B (X), let W ϕ = B %

ϕ, ₂₄ ¹ α(ϕ)

∩ 

ψ ∈ USCC B (X)   ²

3 α(ϕ) < α(ψ) < ⁴ ₃ α(ϕ) , and subdivide K so that each simplex is contained in some q ⁻¹ (W _ϕ ).

For each v ∈ K ⁽⁰⁾ , we define f (v) ∈ USCC B (X) as follows:

f (v) = q(v) ∪ [

a∈D

B a, ¹ ₈ αq(v)

× [b(v, a), t(v, a)],

where

b(v, a) = inf q(v) B a, ¹ ₈ αq(v)

, t(v, a) = sup q(v) B a, ¹ ₈ αq(v)

. Obviously % H (f (v), q(v)) ≤ ¹ ₈ αq(v). If u and v are vertices of the same simplex of K, then

% _H (f (u), f (v)) ≤ % _H (f (u), q(u)) + % _H (q(u), q(v)) + % _H (f (v), q(v))

< ¹ ₈ αq(u) + ¹ ₈ αq(v) + ¹ ₈ αq(v) < ¹ ₄ αq(v) + ¹ ₄ αq(v) = ¹ ₂ αq(v).

For the barycenter b σ of each σ ∈ K, we define f (b σ) ∈ USCC B (X) by f (b σ)(x) =

h

min [

v∈σ

f (v)(x), max [

v∈σ

f (v)(x) i

.

Then, by Lemma 2.1, f (b σ) ⊂ N _% f (v), ¹ ₂ αq(v)

for each v ∈ σ ⁽⁰⁾ . Observe that if 0 < r ≤ min _v∈σ

1

8 αq(v), then

f (b σ)|B(a, r) = B(a, r) × [b(b σ, a), t(b σ, a)] for each a ∈ D, where b(b σ, a) = min _v∈σ

b(v, a) and t(b σ, a) = max _v∈σ

t(v, a).

We define a map f : |K| → USCC B (X) as follows:

f (y)(x) = X k i=1

s i f (b σ i )(x) = h X ^k

i=1

s i min f (b σ i )(x), X k i=1

s i max f (b σ i )(x) i

,

where y = P _k

i=1 s _i b σ _i , σ ₁ < . . . < σ _k ∈ K, s _i ≥ 0 and P _k

i=1 s _i = 1. In the above, note that ¹ ₂ αq(y) < αq(v) for each v ∈ σ ⁽⁰⁾ _k . Then, for each a ∈ D,

f (y)|B a, ₁₆ ¹ αq(y)

= B a, ₁₆ ¹ αq(y)

× [min f (y)(a), max f (y)(a)].

For each y ∈ |K|, choose v ∈ σ ⁽⁰⁾ so that y ∈ |St(v, Sd K)|. Since f (v) ⊂ f (y) ⊂ f (b σ) ⊂ N % f (v), ¹ ₂ αq(v)

, we have % H (f (y), f (v)) < ¹ ₂ αq(v), hence

% H (f (y), q(y)) ≤ % H (f (y), f (v)) + % H (f (v), q(v)) + % H (q(v), q(y))

< ¹ ₂ αq(v) + ¹ ₈ αq(v) + ¹ ₈ αq(v) < ³ ₄ αq(v) < ³ ₂ αq(y).

Now, for any F ∈ 2 ^D , we define h _F : USCC _B (X) → USCC _B (X) by h _F (ϕ) = f p(ϕ) ∪ [

a∈F

{a} × 

max f p(ϕ)(a), max f p(ϕ)(a) + ¹ ₂ αqp(ϕ)  . Then h _F is U-close to id. In fact, h _F is V-close to qp because

% _H (h _F (ϕ), qp(ϕ)) ≤ % _H (h _F (ϕ), f p(ϕ)) + % _H (f p(ϕ), qp(ϕ))

< ¹ ₂ αqp(ϕ) + ³ ₂ αqp(ϕ) = 2αqp(ϕ).

To show the continuity of h F , let ϕ n → ϕ in USCC B (X) as n → ∞. Let 0 < r < ₁₆ ¹ αqp(ϕ). Since αqp is continuous, r < ₁₆ ¹ αqp(ϕ _n ) for sufficiently large n, whence for each a ∈ F ,

f p(ϕ n )|B(a, r) = B(a, r) × [min f p(ϕ n )(a), max f p(ϕ n )(a)] and f p(ϕ)|B(a, r) = B(a, r) × [min f p(ϕ)(a), max f p(ϕ)(a)].

On the other hand, f p(ϕ _n ) → f p(ϕ) because f p is continuous. Then, as is easily observed, max f p(ϕ n )(a) → max f p(ϕ)(a) for each a ∈ F . From the definition, it follows that h _F (ϕ _n ) → h _F (ϕ).

We show that {h _F (USCC _B (X)) | F ∈ 2 ^D } is discrete in USCC _B (X).

Suppose that, on the contrary, there exist ϕ, ϕ i ∈ USCC B (X) and F i ∈ 2 ^D

(i ∈ N) such that h _F

(ϕ _i ) → ϕ as i → ∞ and F _i 6= F _j if i 6= j. Then

inf i∈N αqp(ϕ i ) > 0. Otherwise, lim n→∞ αqp(ϕ i

) → 0 for some i 1 < i 2 < . . . As seen above, % _H (h _F

(ϕ _i

), qp(ϕ _i

)) < 2αqp(ϕ _i

). Then it follows that qp(ϕ _i

) → ϕ, hence αqp(ϕ) = lim _n→∞ αqp(ϕ _i

) = 0, which is a contradic- tion.

Let ε ₀ = inf _{i∈N 16} ¹ αqp(ϕ _i ) > 0. For any i 6= j ∈ N, there exists a ∈ D such that a ∈ F _i \F _j or a ∈ F _j \F _i . Without loss of generality, we may assume that a ∈ F j \ F i . For simplicity, we write b i = b(p(ϕ i ), a), t i = t(p(ϕ i ), a), b _j = b(p(ϕ _j ), a) and t _j = t(p(ϕ _j ), a). Then

h F

(ϕ i )|B(a, ε 0 ) = B(a, ε 0 ) × [b i , t i ] and

h _F

(ϕ _j )|B(a, ε ₀ ) = B(a, ε ₀ ) × [b _j , t _j ] ∪ {a} × [t _j , t _j + αqp(ϕ _j )].

In case t _i ≤ t _j + ¹ ₂ αqp(ϕ _j ), we have

% H (h F

, h F

) ≥ %((a, t j + αqp(ϕ j )), h F

) ≥ min 

ε 0 , ¹ ₂ αqp(ϕ j )

= ε 0 . Recall that a has a connected neighborhood with diameter > δ. Since ε ₀ <

1

16 δ, there is c ∈ X so that d(a, c) = ε 0 /2. In case t i ≥ t j + ¹ ₂ αqp(ϕ j ), it follows that

% H (h F

, h F

) ≥ %((c, t i ), h F

) ≥ min 

ε 0 /2, ¹ ₂ αqp(ϕ j )

= ε 0 /2.

Consequently, % _H (h _F

(ϕ _i ), h _F

(ϕ _j )) ≥ ε ₀ /2 if i 6= j, whence h _F

(ϕ _i ) is not convergent. This is a contradiction.

3.3. Lemma. Assume that X is not totally bounded. For each n ∈ N, let D n be a maximal subset of X such that d(x, y) ≥ 2 ⁻ⁿ for any distinct points x, y ∈ D _n ( ⁴ ). Then w(USCC _B (X)) = sup _n∈N 2 ^{card D}

. In case X is separable, w(USCC _B (X)) = 2 ^ℵ

( ⁵ ).

P r o o f. For each n ∈ N, let Q _n = {2 ⁻ⁿ m | m ∈ N} ⊂ R. Then D _n × Q _n is discrete in X × R. Since X is not totally bounded, each D _n is infinite, hence card(D n × Q n ) = card D n . By the maximality, d(x, D n ) < 2 ⁻ⁿ for every x ∈ X, hence %(z, D _n × Q _n ) < 2 ⁻ⁿ for every z ∈ X × R. For each E ∈ 2 ^X×R and n ∈ N, let

F = {z ∈ D _n × Q _n | %(z, E) < 2 ⁻ⁿ } ∈ 2 ^D

^×Q

⊂ 2 ^X×R . Then % _H (E, F ) ≤ 2 ⁻ⁿ . Hence, S

n∈N 2 ^D

^×Q

is dense in 2 ^X×R . Since the weight w(2 ^X×R ) is equal to the density of 2 ^X×R , it follows that

w(2 ^X×R ) ≤ card [

n∈N

2 ^D

^×Q

≤ sup

n∈N

card 2 ^D

^×Q

= sup

n∈N

2 ^card(D

^×Q

⁾ = sup

n∈N

2 ^{card D}

, which implies w(USCC _B (X)) ≤ sup _n∈N 2 ^{card D}

.

(

) The existence of such D

⊂ X is guaranteed by Zorn’s Lemma.

(

) In general, sup

2

6= 2

= 2

.

On the other hand, for each n ∈ N and F ∈ 2 ^D

, let ϕ _F = F × I ∪ X × {0} ∈ USCC _B (X).

Since % _H (ϕ _F , ϕ _F

) ≥ 2 ⁻ⁿ for each F 6= F ⁰ ∈ 2 ^D

, {B _%

(ϕ _F , 2 ⁻ⁿ⁻¹ ) | F ∈ 2 ^D

} is pairwise disjoint. Therefore, w(USCC B (X)) ≥ card 2 ^D

= 2 ^{card D}

, hence w(USCC _B (X)) ≥ sup _n∈N 2 ^{card D}

.

Proof of Main Theorem. We apply Lemma 3.1 to show that USCC _B (X) ≈

` ₂ (A), where card A = w(USCC _B (X)). We have proved that USCC _B (X) is a completely metrizable AR (Corollary 1.3 and Theorem 2.2). It remains to construct a map f : USCC _B (X) × A → USCC _B (X) such as in Lemma 3.1.

Let C be the collection of all components of X and take D n (n ∈ N) as in Lemma 3.3. Then observe that

card C ≤ w(X) = card [

n∈N

D _n = sup

n∈N

card D _n .

Case (1): card C = w(X). Since X is uniformly locally connected, card D _n

≥ card C = w(X) for sufficiently large n ∈ N. On the other hand, card D _n ≤ w(X) for all n ∈ N by definition. Then sup _n∈N 2 ^{card D}

= 2 ^w(X) , hence Lemma 3.3 yields card A = w(USCC _B (X)) = 2 ^w(X) .

We can write C = S

i∈N C _i , where C _i ∩ C _j = ∅ if i 6= j and card C _i = w(X) for each i ∈ N. For each i ∈ N, let r _i : USCC _B (X) → m(C _i ) be the map defined by r i (ϕ)(C) = sup ϕ(C) (≤ sup ϕ(X)) for each C ∈ C i . Since m(C _i ) ≈ ` ₂ (2 ^C

) ([BP, Ch. VII, Theorem 6.1]) and w(USCC _B (X) × A) = 2 ^w(X) = card 2 ^C

, there is a closed embedding g i : USCC B (X) × A → m(C i ) such that kg _i (ϕ, a) − r _i (ϕ)k < 2 ⁻ⁱ for each (ϕ, a) ∈ USCC _B (X) × A. Note that {(g _i ) _a (USCC _B (X)) | a ∈ A} is discrete in USCC _B (X).

For any open cover U of USCC _B (X), let α : USCC _B (X) → (0, 1) be a map such that {B %

(ϕ, α(ϕ)) | ϕ ∈ USCC B (X)} ≺ U. Now, we define a map f : USCC _B (X) × A → USCC _B (X) as follows:

f (ϕ, a)(x) =

 

 

 

 

 

 

 



ϕ(x) + g _i (ϕ, a)(C) − r _i (ϕ)(C)

for x ∈ C ∈ C _i and 2 ⁻ⁱ⁺¹ < α(ϕ), ϕ(x) + 2 ⁱ (α(ϕ) − 2 ⁻ⁱ )(g _i (ϕ, a)(C) − r _i (ϕ)(C))

for x ∈ C ∈ C i and 2 ⁻ⁱ ≤ α(ϕ) ≤ 2 ⁻ⁱ⁺¹ , ϕ(x) otherwise.

Then f _a is U-close to id. In fact, for every C ∈ C _i ,

|g i (ϕ, a)(C) − r i (ϕ)(C)| ≤ kg i (ϕ, a) − r i (ϕ)k < 2 ⁻ⁱ , hence % _H (f _a (ϕ), ϕ) < α(ϕ).

We prove that {f _a (USCC _B (X)) | a ∈ A} is discrete in USCC _B (X).

Suppose that, on the contrary, there is a sequence (ϕ k , a k ) ∈ USCC B (X)×A

(k ∈ N) such that a _k 6= a _k

if k 6= k ⁰ , and f _a

(ϕ _k ) converges to some

ϕ 0 ∈ USCC B (X). Then there is some i 0 ∈ N such that 2 ⁻ⁱ

⁺¹ < α(ϕ k ) for all k ∈ N. Otherwise, lim _j→∞ α(ϕ _k(j) ) = 0 for some k(1) < k(2) < . . . , whence lim _j→∞ % _H (f _a

(ϕ _k(j) ), ϕ _k(j) ) = 0. It follows that ϕ _k(j) converges to ϕ 0 , so α(ϕ 0 ) = lim j→∞ α(ϕ _k(j) ) = 0, which is a contradiction. For each C ∈ C _i

,

r _i

(f _a

(ϕ _k ))(C) = sup f (ϕ _k , a _k )(C)

= sup ϕ k (C) + g i

(ϕ k , a k )(C) − r i

(ϕ k )(C)

= g _i

(ϕ _k , a _k )(C) = (g _i

) _a

(ϕ _k ).

Since r _i

is continuous, (g _i

) _a

(ϕ _k ) = r _i

(f _a

(ϕ _k )) converges to r _i

(ϕ ₀ ), which contradicts the fact that {(g i

) a (USCC B (X)) | a ∈ A} is discrete in USCC _B (X). Therefore, {f _a (USCC _B (X)) | a ∈ A} is discrete in USCC _B (X).

Case (2): card C < w(X). Since X is uniformly locally connected, we may assume the condition (∗) of §2. Let X ₀ be the set of isolated points of X. Then d(x, X \ {x}) ≥ 1 for every x ∈ X ₀ by (∗). As is easily seen,

USCC B (X) ≈ USCC B (X 0 ) × USCC(X \ X 0 ).

For each n ∈ N, let D ⁰ _n = D n \ X 0 . Since card X 0 ≤ card C < w(X) = sup _n∈N card D _n , we have card X ₀ < card D _n for sufficiently large n ∈ N, whence card D ⁰ _n = card D n . By Lemma 3.3,

w(USCC B (X \ X 0 )) = sup

n∈N

2 ^{card D}

= sup

n∈N

2 ^{card D}

= w(USCC B (X)).

In case (1) above, we have shown that USCC _B (X ₀ ) is homeomorphic to a Hilbert space, hence it is a completely metrizable AR with

w(USCC B (X 0 )) ≤ w(USCC B (X)).

By [To 2 , Theorem 3.1], it suffices to show that USCC B (X \ X 0 ) is homeo- morphic to a Hilbert space with the same weight. Thus we can assume that X has no isolated points.

For each δ > 0, let C(δ) = {C ∈ C | diam C < δ}. Let D ¹ _n = D n \ [

C(2 ⁻ⁿ ) for each n ∈ N.

Note that each point of D ¹ _n has a connected neighborhood in X with diam ≥ 2 ⁻ⁿ because it is contained in a component of X with diam ≥ 2 ⁻ⁿ . Each member of C(2 ⁻ⁿ ) contains at most one point of D _n . Recall that card C <

w(X) = sup _n∈N card D _n . Then, for sufficiently large n ∈ N, card



D _n ∩ [

C(2 ⁻ⁿ )



≤ card C(2 ⁻ⁿ ) ≤ card C < card D _n ,

whence card D n = card D _n ¹ . Therefore, it follows from Lemma 3.3 that card  [

n∈N

{n} × 2 ^D



= sup

n∈N

card 2 ^D

= sup

n∈N

2 ^{card D}

= sup

n∈N

2 ^{card D}

= w(USCC _B (X)).

Thus we may assume that

A = [

n∈N

{n} × 2 ^D

.

For any open cover U of USCC B (X), let V be an open star-refinement of U. Since X is not totally bounded, we can apply Lemma 3.2 to obtain a map g : USCC _B (X) × N → USCC _B (X) such that {g _n (USCC _B (X)) | n ∈ N}

is discrete in USCC B (X) and each g n is V-close to id. Choose an open refinement W of V so that the star st(W, W) of each W ∈ W meets at most one of g n (USCC B (X)). Applying Lemma 3.2 again, we obtain maps h n : USCC B (X) × 2 ^D

→ USCC B (X) (n ∈ N) such that {(h n ) F (USCC B (X)) | F ∈ 2 ^D

} is discrete in USCC B (X) and each (h n ) F is W-close to id. Then we define a map f : USCC _B (X) × A → USCC _B (X) by

f (ϕ, (n, F )) = h _n (g(ϕ, n), F ) (i.e., f _{(n,F )} (ϕ) = (h _n ) _F ◦g _n (ϕ)).

Each f _{(n,F )} is U-close to id because it is W-close to g _n .

We show that the collection {f _{(n,F )} (USCC _B (X)) | (n, F ) ∈ A} is discrete in USCC B (X). Each ϕ ∈ USCC B (X) is contained in some W ∈ W. Then this W meets at most one member of {f (USCC _B (X) × {n} × 2 ^D

) | n ∈ N}.

In fact, if f _{(n,F )} (ψ), f _(n

,F

) (ψ ⁰ ) ∈ W for some ψ, ψ ⁰ ∈ USCC _B (X), n 6=

n ⁰ ∈ N, F ∈ 2 ^D

and F ⁰ ∈ 2 ^D

, then g n (ψ), g n

(ψ ⁰ ) ∈ st(W, W), which is a contradiction. In case

W ∩ f (USCC _B (X) × {n} × 2 ^D

) 6= ∅,

we can choose a neighborhood W ⁰ of ϕ so that W ⁰ ⊂ W and W ⁰ meets at most one of (h _n ) _F (USCC _B (X)). Since

f _{(n,F )} (USCC _B (X)) = (h _n ) _F ◦g _n (USCC _B (X)) ⊂ (h _n ) _F (USCC _B (X)), W ⁰ meets at most one of f _{(n,F )} (USCC _B (X)). Thus {f _{(n,F )} (USCC _B (X)) | (n, F ) ∈ A} is discrete in USCC _B (X).

Finally, we show that USCC(X, [−1, 1]) ≈ ` 2 (A) (i.e., USCC(X, I) ≈

` ₂ (A)). Let

B = {ϕ ∈ USCC(X, [−1, 1]) | inf ϕ(X) = −1 or sup ϕ(X) = 1}.

Then B is clearly closed in USCC(X, [−1, 1]) and

USCC(X, [−1, 1]) \ B ≈ USCC _B (X) ≈ ` ₂ (A).

We show that B is a strong Z-set in USCC(X, [−1, 1]), whence we obtain USCC(X, [−1, 1]) ≈ ` ₂ (A) by [To ₄ , Theorem B1] (cf. [To ₂ ]). For any map α : USCC(X, [−1, 1]) → (0, 1), we define a map

h : USCC(X, [−1, 1]) → USCC(X, [−1, 1]) by h(ϕ)(x) = 1 − ¹ ₂ α(ϕ)

· ϕ(x). Then % _H (h(ϕ), ϕ) < α(ϕ) for each ϕ ∈ USCC(X, [−1, 1]). For every ϕ 0 ∈ cl h(USCC(X, [−1, 1])), there is a se- quence ϕ _k ∈ USCC(X, I) (k ∈ N) such that h(ϕ _k ) → ϕ ₀ . Then b = inf _k∈N α(ϕ _k ) > 0. Otherwise, lim _j→∞ α(ϕ _k

) = 0 for some k ₁ < k ₂ < . . . , hence ϕ k

converges to ϕ 0 , so α(ϕ 0 ) = lim j→∞ α(ϕ k

) = 0, which is a con- tradiction. For each k ∈ N,

sup [

x∈X

h(ϕ _k )(x) =

 1 − 1

2 α(ϕ _k )



· sup [

x∈X

ϕ _k (x) ≤ 1 − 1 2 b, hence sup S

x∈X ϕ ₀ (x) ≤ 1 − ¹ ₂ b < 1. Similarly, we have inf S

x∈X ϕ ₀ (x) ≥

−1 + ¹ ₂ b > −1. Therefore, ϕ 0 6∈ B. This means that B ∩ cl h(USCC(X, [−1, 1])) = ∅.

Thus B is a strong Z-set in USCC(X, [−1, 1]).

Remark. Let P be the convex set in the Banach space C _B (X) ² = C _B (X) × C _B (X) defined as follows:

P = {(f, g) ∈ C _B (X) ² | g(x) ≥ 0 for all x ∈ X}.

Then it is easy to see that if X = (X, d) is a discrete metric space (i.e., inf{d(x, y) | x 6= y} > 0), then USCC B (X) ≈ P . In fact, for each ϕ ∈ USCC _B (X), we define m _ϕ , r _ϕ ∈ C _B (X) by

m ϕ (x) = ¹ ₂ (min ϕ(x) + max ϕ(x)), r _ϕ (x) = ¹ ₂ (max ϕ(x) − min ϕ(x)).

Then the desired homeomorphism ξ : USCC _B (X) → P can be defined by ξ(ϕ) = (m ϕ , r ϕ ).

4. Remarks on topologies for C _B (X) and C(X, I). Although the

spaces C _B (X) and C(X, I) with the sup-metric are AR’s for an arbitrary

metric space X, the example in the Introduction also shows that the spaces

C _B (X) and C(X, I) with the Hausdorff metric % _H are not ANR’s even if X is

locally connected. One should also remark that C _B (X) is not a topological

linear space in this topology. In fact, it can easily be derived from [FK,

Remark 3.6] that the addition C _B (R) ² → C _B (R) ((f, g) 7→ f + g) is not

continuous with respect to the Hausdorff metric. However, we can prove the

following:

4.1. Theorem. For any uniformly locally connected metric space X = (X, d), the spaces C _B (X) and C(X, I) with the Hausdorff metric are AR’s.

A subset Z of a space Y is said to be homotopy dense in Y if there exists a homotopy h : Y × I → Y such that h ₀ = id and h _t (Y ) ⊂ Z for t > 0.

As is easily observed, a homotopy dense subset of an AR (resp. ANR) is also an AR (resp. ANR). By Theorem 2.2, in case X has no isolated points, Theorem 4.1 is deduced from the following:

4.2. Theorem. For any uniformly locally connected metric space X = (X, d) with no isolated points, C B (X) (resp. C(X, I)) is homotopy dense in USCC _B (X) (resp. USCC(X, I)).

As a corollary of Theorem 4.2, we also have the following:

4.3. Corollary. Let X = (X, d) be an infinite σ-compact complete metric space, which is assumed to be uniformly locally connected in case X is non-compact. Then C B (X) and C(X, I) with the Hausdorff metric are homeomorphic to a Hilbert space.

To prove Theorem 4.2, we need the following non-compact version of [SU, Lemma 2]:

4.4. Lemma. Assume that condition (∗) of §2 holds and X has no isolated points. Let f 0 : K ⁽⁰⁾ → C B (X) be a map of the 0-skeleton of a locally finite simplicial complex K such that diam _%

f ₀ (σ ⁽⁰⁾ ) < 1 for every σ ∈ K, where σ ⁽⁰⁾ = σ ∩ K ⁽⁰⁾ . Then f ₀ extends to a map f : |K| → C _B (X) such that

diam _%

f (σ) ≤ 4 diam _%

f ₀ (σ ⁽⁰⁾ ) for every σ ∈ K, where C _B (X) has the topology induced by % _H .

S k e t c h o f p r o o f. By Lemma 2.1, we have property (]). Then the proof is the same as that of [SU, Lemma 2], with C(X, (−1, 1)) replaced by C B (X). Now, since X is not compact, we cannot take A v ⊂ X as a finite set in the proof, but since K is locally finite and X has no isolated points, we can take A v ⊂ X as a discrete set with the same property, that is, f (v) ⊂ N _% (f (v)|A _v , ε _v ) (in other words, f (v)|A _v = f (v) ∩ p ⁻¹ (A _v ) is ε _v -dense in f (v)), and each A _v has an open neighborhood U _v in X with U v ∩ U v

= ∅ if v 6= v ⁰ ∈ σ ⁽⁰⁾ and σ ∈ K. No other change is necessary.

Remark. In the above, if card St(v 0 , K) > card X at some vertex v 0 ∈ K ⁽⁰⁾ , it is impossible to obtain discrete sets A _v ⊂ X, v ∈ K ⁽⁰⁾ , such that A _v ∩ A _v

= ∅ for every v ∈ St(v ₀ , K) ⁽⁰⁾ . Then the local finiteness of K is assumed.

We can apply Lemma 4.4 to prove the following result the same way as

[SU, Lemma 3].

4.5. Lemma. Let X = (X, d) be a uniformly locally connected metric space with no isolated points and f : Y → USCC _B (X) a map of a separable metrizable space Y . Then there exists a homotopy h : Y × I → USCC _B (X) such that h 0 = f and h t (Y ) ⊂ C B (X) for t > 0.

P r o o f. By replacing the metric d by d c , we can assume condition (∗) of

§2. For each n ∈ N, let U _n be an open cover of USCC _B (X) with mesh _%

U _n <

(n + 1) ⁻¹ . Since Y is separable metrizable, the open cover f ⁻¹ (U _n ) of Y has a countable star-finite open refinement V n , whence the nerve of V n is locally finite. We define

W ₁ = {U × (2 ⁻¹ , 1] | U ∈ U ₁ },

W n = {U × ((n + 1) ⁻¹ , (n − 1) ⁻¹ ) | U ∈ U n } for n > 1.

Thus we have a star-finite open cover W = S

n∈N W n of Y × (0, 1]. Let K be the nerve of W and g : Y × (0, 1] → |K| a canonical map, that is, each g(y, t) is contained in the simplex spanned by all vertices W ∈ W containing (y, t). Then K is locally finite. For each n ∈ N, let K n be the nerve of W _n ∪ W _n+1 . Then each K _n is a subcomplex of K and K = S

n∈N K _n . Note that K ⁽⁰⁾ = S

n∈N W _n . For each W ∈ W _n , since pr _Y (W ) ∈ V _n ≺ f ⁻¹ (U _n ), we can choose π(W ) ∈ U n so that f pr _Y (W ) ⊂ π(W ).

Since C B (X) is dense in USCC B (X) by Theorem 1.5, we can also choose k ₀ (W ) ∈ π(W ) ∩ C _B (X), whence % _H (k ₀ (W ), f (y)) ≤ mesh _%

U _n < (n + 1) ⁻¹ for any y ∈ pr _Y (W ). Thus we have a map k ₀ : K ⁽⁰⁾ → C _B (X) such that

% _H (k ₀ (W ), f (y)) < (n + 1) ⁻¹ for any W ∈ K n ⁽⁰⁾ = W _n and y ∈ pr _Y (W ), hence diam %

k 0 (σ ⁽⁰⁾ ) < 2(n + 1) ⁻¹ for each σ ∈ K n . By using Lemma 4.4, we can extend k ₀ to a map k : |K| → C _B (X) such that diam _%

k(σ) <

4 diam %

k 0 (σ ⁽⁰⁾ ). Thus we obtain the map

kg : Y × (0, 1] → C B (X) ⊂ USCC B (X).

For each (y, t) ∈ Y ×(0, 1], choose n ∈ N and W ∈ W _n so that (n+1) ⁻¹ <

t ≤ n ⁻¹ and (y, t) ∈ W . Then there is σ ∈ K n such that g(y, t) ∈ σ and W ∈ σ ⁽⁰⁾ . Since k(W ), kg(y, t) ∈ k(σ) and diam _%

k(σ) < 4 diam _%

k(σ ⁽⁰⁾ ) <

8(n + 1) ⁻¹ , it follows that

% H (kg(y, t), f (y)) ≤ % H (kg(y, t), k(W )) + % H (k(W ), f (y))

< 8(n + 1) ⁻¹ + (n + 1) ⁻¹ = 9(n + 1) ⁻¹ < 9t.

Then kg can be extended to the desired homotopy h by h ₀ = f .

Remark. In the above lemma, the separability of Y is necessary because the local finiteness of K is assumed in Lemma 4.4. Note that USCC _B (X) is non-separable.

A subset Z ⊂ Y is called locally homotopy negligible in Y if every neigh-

borhood U of each point x ∈ X contains a neighborhood V of x such that