THE HYPERSPACE OF THE REGIONS BELOW OF

Questions and Answers in General Topology 2x (200x), pp. xxx–yyy

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THE HYPERSPACE OF THE REGIONS BELOW OF CONTINUOUS MAPS FROM S × S TO I ZHONGQIANG YANG AND NADA WU (Communicated by Jun-iti Nagata)

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Abstract. Let S = {1, 12 , 212 , · · · , 0} be the converging sequence with the limit. Let T = S × S be the product space and let T ′ be the subspace consisting of all limit points in T . We denote, by USC(T ) and C(T ), the families of all upper semi-continuous functions and continuous functions from T to I = [0, 1], respectively. Let C0 (T ) = {f ∈ C(T ) : f (a) = 0 for all a ∈ T ′ }. For each f ∈ USC(T ) we define the closed subset ↓ f = {(x, t) ∈ T × I : t ≤ f (x)} of the product space T × I. Then ↓ USC(T ) = {↓ f : f ∈ USC(T )} may be considered as a subspace of the hyperspace of all nonempty closed subsets of X × I endowed with the Vietoris topology. ↓ C(T ) = {↓ f : f ∈ C(T )} and ↓ C0 (T ) = {↓ f : f ∈ C0 (T )} are its subspaces. In this paper we will show that there exist two homeomorphisms h1 : (↓USC(T ), ↓C0 (T )) ∼ = (Q, s), where Q = [−1, 1]ω is the Hilbert cube and s = (−1, 1)ω is the pseudo interior, and h2 : (↓USC(T ), ↓C(T )\ ↓C0 (T )) ∼ = (Q, c0 ), where c0 = {(xn ) ∈ Q : lim xn = 0}.

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However, we do not know any topological model of ↓C(T ).

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1. Introduction

For a Tychonoff space X, the hyperspace Cld(X) is the set consisting of all nonempty closed subsets in X endowed with the Vietoris topology which is generated by the subbase {U − , U + : U ⊂ X is open}, where U − = {A ∈ Cld(X) : A ∩ U ̸= ∅} and U + = {A ∈ Cld(X) : A ⊂ U }.

It is well-known that Cld(X) is metrizable if and only if X is compact and metrizable [8, Theorem I.3.4]. For a compact metric space X = (X, d), the Vietoris topology of Cld(X) is induced by the Hausdorff metric dH defined as follows: dH (E, F ) = inf{ε > 0 : E ⊂ B(F, ε) and F ⊂ B(E, ε)}. 2000 Mathematics Subject Classification. 54B20, 57N20, 54E45. Key words and phrases. Regions below; Upper Semi-continuous; The Hilbert Cube; Pseudointerior; Strongly universal; The converging sequence; The capset. This work was supported by Nation Natural Science Foundation of China (No. 10471084) . 1

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Zhongqiang Yang and Nada Wu

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A celebrated result is Curtis-Schori-West Hyperspace Theorem [6] which asserts that Cld(X) is homeomorphic to (∼ =) the Hilbert cube Q = [−1, 1]ω if and only if X is a non-degenerate connected, locally connected and compact metric space. For a Tychonoff space X and a subset L of the set R of all real numbers, we consider the sets C(X, L) and USC(X, L) which consist of all continuous maps and all upper semi-continuous maps from X to L, respectively. For f ∈ USC(X, L), let ↓f = {(x, λ) ∈ X × L : λ ≤ f (x)}. Moreover, for a subset A of USC(X, L), let ↓A = {↓f : f ∈ A}. Note that ↓f is closed in X × L for each f ∈ USC(X, L). Thus, as subspaces of Cld(X × L), ↓USC(X, L) and ↓C(X, L) are topological spaces. It is not hard to verify that χ : Cld(X) ⊕ {∅} →↓USC(X, {0, 1}) is a homeomorphism, where { 1 if x ∈ A χ(A)(x) = 0 if x ̸∈ A

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for each A ∈ Cld(X) ⊕ {∅} and x ∈ X. Thus, Curtis-Schori-West Hyperspace Theorem may be rewritten that ↓USC(X, {0, 1}) \ {↓χ(∅)} ∼ = Q if and only if X is a non-degenerate connected, locally connected and compact metric space. In [12-15], the case L = I = [0, 1] was considered. Then USC(X, I) and C(X, I) are abbreviated to USC(X) and C(X), respectively. In those papers, the following theorems were proved:

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Theorem A ([14, Theorem 1]). For an infinite Tychonoff space X, the following conditions are equivalent: (a) X is a compact metric space; (b) ↓C(X) has a countable base; (c) ↓USC(X) ∼ = Q.

Theorem B ([14, Theorem 3]). For a Tychonoff space X, the following conditions are equivalent: (a) X is a compact metric space and the set of isolated points is not dense; (b) ↓C(X) ∼ = c0 , where c0 = {(xn ) ∈ Q : lim xn = 0}; n→∞

(c) there exists a homeomorphism h :↓USC(X) → Q such that h(↓C(X)) = c0 , that is, (↓USC(X), ↓C(X)) ∼ = (Q, co ).

However for a compact metric space X whose isolated points are dense, even for the case X = S = {1, 21 , 212 , · · · , 0}, we do not know any topological model of ↓C(X). In [15], They showed that if a compact metric space X has a dense subset of isolated points, ↓C(X) is homeomorphic to neither c0 nor (−1, 1)ω . Thus, in this case the structure of ↓C(X) is complicated. On the other hand, they showed the following theorem in [15]. Here C0 (S) = {f ∈ C(S) : f (0) = 0}.

The Hyperspace of the regions below of continuous maps from S × S to I

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Theorem C ([15, Theorems 1 and 2]). (↓USC(S), ↓C0 (S)) ∼ = (Q, s), where s = ω ∼ (−1, 1) , and (↓USC(S), ↓C(S)\ ↓C0 (S)) = (Q, c0 ).

Theorem 1. (↓USC(T ), ↓C0 (T )) ∼ = (Q, s).

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In the present paper, we consider the case that 1 1 X = T = S × S = {( i , j ) : i, j ∈ N ∪ {∞}} 2 2 and try to find out the similarity between ↓C(T ) and ↓C(S). Let T ′ = {( 21i , 21j ) : i, j ∈ N ∪ {∞} and either i = ∞ or j = ∞} and C0 (T ) = {f ∈ C(T ) : f (a) = 0 for all a ∈ T ′ }. We have the following theorems.

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Theorem 2. (↓USC(T ), ↓C(T )\ ↓C0 (T )) ∼ = (Q, c0 ). However, the following problems remain open.

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Problem 1. What is (↓USC(T ), ↓C(T )) homeomorphic to? More generally, what is (↓USC(X), ↓C(X)) homeomorphic to for any infinite compact metric space X with dense set of isolated points? Problem 2. Whether

(↓USC(X), ↓C0 (X)) ∼ = (Q, s) and (↓USC(X), ↓C(T )\ ↓C0 (X)) ∼ = (Q, c0 )

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for any infinite compact metric space with dense set of isolated points? 2. Preliminaries

All spaces under discussion are assumed to be separable metrizable.

Definition 1. Let X be a space. A (single-valued) function f : X → I is called upper semi-continuous (abbreviated USC) if f −1 [0, t) is open in X for every t ∈ I. Definition 2. A closed subset A of a metric space (X, d) is said to be a Z-set of X (denoted by A ∈ Z(X)) if for every continuous map ε : X → (0, +∞) there is a continuous map f : X → X\A with d(f (x), x) < ε(x) (x ∈ X). If X is compact, then the above map ε can be replaced by an arbitrary positive real number ε. A σZ-set of X is a countable union of Z-sets (If A is a σZ-set of X, we denote it by A ∈ Zσ (X)). A Z-embedding is an embedding with a Z-set image. Definition 3. Let (M Q , d) be a copy of the Hilbert cube Q. An element A ∈ Zσ (M Q ) is called a capset provided that A can be written as the union of an increasing sequence A1 ⊂ A2 ⊂ ... ⊂ An ⊂ ... of Z-sets in M Q and the following absorption properties hold: For any ε > 0, any n ∈ N, Z ∈ Z(M Q ) there exist m > n and a homeomorphism h : M Q → M Q such that

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(a) d(h, id) < ε; (b) h|An = id; (c) h(Z) ⊂ Am . Definition 4. We say a metric space (X, d) has the disjoint-cells property, if for any ε > 0, k ∈ N and continuous functions f, g : Ik → X, there exist continuous functions f ′ , g ′ : Ik → X such that d(f ′ , f ) < ε, d(g ′ , g) < ε and f ′ (Ik ) ∩ g ′ (Ik ) = ∅.

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let M0 denote the class of compact spaces, and for a topological class C, let (M0 , C) denote the class of pairs (Z, C) such that Z ∈ M0 , C ∈ C and C ⊂ Z.

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Definition 5. We say that a pair (X, Y ) ∈ (M0 , C) is strongly (M0 , C)-universal provided that for any (Z, C) ∈ (M0 , C), any map f : Z → X, any closed subset K of Z such that f |K : K → X is a Z-embedding and any ε > 0, there is a Zembedding g : Z → X such that g|K = f |K , g −1 (Y )\K = C\K and d(g(z), f (z)) < ε for all z ∈ Z. Definition 6. We say that a subset Y of X is a C-absorber in X if (a) Y ∈ C. (b) Y is contained in a σZ-set of X, and (c) (X, Y ) ∈ (M0 , C) is strongly (M0 , C)-universal.

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Lemma 1 ([2, Theorem 8.2](cf. [4])). If X and Y are C-absorbers in a Q-manifold M , then (M, X) ∼ = (M, Y ). The above lemma says that absorbers in a Q-manifold are topologically unique and is our main tool to show Theorem 2. The following lemma provides a powerful tool to determine that a given space is homeomorphic to the Hilbert cube Q. Lemma 2 ((Torun´ nczyk’s Characterization Theorem )[11] (cf. [9, Corollary 7.8.4])). A space X is homeomorphic to the Hilbert cube Q if and only if it is a compact AR with the disjoint-cells property. 3. Proof of Theorem 1

In the following, if φ : A → B be a map, where A ⊂ USC(X) and/or B ⊂ USC(Y ) for spaces X and Y . We may define a corresponding map ↓φ : ↓A →↓B or ↓ φ : A →↓ B or ↓ φ : ↓ A → B as ↓ φ(↓ f ) =↓ (φ(f )) or ↓ φ(f ) =↓ (φ(f )) or ↓φ(↓f ) = φ(f ), respectively. Let M Q be a copy of the Hilbert cube Q, then B = (Bi )i be a tower of subsets of M Q , that is, Bi ⊂ Bi+1 ⊂ M Q for every i. We say that B has the def ormation property if there is a homotopy H : M Q × I → M Q such that H0 = idM Q and for each t ∈ (0, 1] there is an i ∈ N such that H(M Q × [t, 1]) ⊂ Bi . Concerning capset and deformation property we know the following lemmas which are the main tools


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to provide a desired homeomorphism. We refer readers to the book [10] for an unified approach to infinite-dimensional manifolds. Lemma 3 ([1](cf. [10, Theorem 5.4.9 ])). An element A ∈ Zσ (Q) is a capset if and only if (Q, A) ∼ = (Q, B(Q)), here B(Q) = Q \ s.

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Lemma 4 ([5](cf. [10, Corollary 5.4.11 ])). An element B ∈ Zσ (Q) is a capset if and only if B can be written as the union of a tower (Bn )n of Z-sets in Q such that: (a) each Bn is homeomorphic to Q; (b) each Bn is a Z-set of Bn+1 , and (c) (Bn )n has the def ormation property.

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To show that (↓ USC(T ), ↓ C0 (T )) ∼ = (Q, s), by Theorem A and Lemma 3, it suffices to prove that ↓USC(T )\ ↓C0 (T ) is a capset in ↓USC(T ) ∼ = Q. For every n ∈ N, let 1 Fn = {↓f ∈↓USC(T ) : there exists a ∈ T ′ such that f (a) ≥ }, n ∪∞ then F = n=1 Fn =↓USC(T )\ ↓USC0 (T ) =↓USC(T )\ ↓C0 (T ), where ↓USC0 (T ) = {↓f ∈↓USC(T ) : f (a) = 0 for any a ∈ T ′ }. For every n ∈ N, let 1 Gn = {↓f ∈↓USC(T ) : f (a) ≥ for any a ∈ T ′ }, n ∪∞ then Gn ⊂ Fn ∩ Gn+1 , and G = n=1 Gn ⊂ F .

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Lemma 5 (cf. [10, Theorem 5.4.3]). Let M Q be a Hilbert cube, and let A, B ∈ Zσ (M Q ) with A a capset. Then A ∪ B is a capset. We will prove that F ∈ Zσ (↓USC(T )) and G is a capset, so that F = F ∪ G is also a capset. We need the following lemmata to check that Fn , Gn ∈ Z(↓USC(T )) and the tower (Gn )n satisfies the conditions in Lemma 4.

Lemma 6. For each n ∈ N, Fn is a closed subset of ↓USC(T ).

Proof. Let A = {A ⊂ T \ T ′ : A is finite}. For any A ∈ A and n ∈ N, let UA,n = {↓f ∈↓USC(T ) : f (x) < n1 for any x ∈ T \ A}. Then UA,n = ((A × I) ∪ ∪ ((T \ A) × [0, n1 )))+ ∩ ↓USC(T ) is open in ↓USC(T ). Let Un = {UA,n : A ∈ A}, then Fn =↓USC(T ) \ Un . Thus Fn is closed in ↓USC(T). ¤ 1 1 For every i ∈ N, let Xi = {( 2n+i , 21i ) : n ∈ N ∪ {∞}}, Yi = {( 21i , 2i+n ) : n ∈ N∪{∞}}, Tn,n = {( 21i , 21j ) : n ≤ i, j ≤ ∞}. We define MXi , MYi and M :↓USC(T )× N → I respectively by, MXi (↓f, n) = max{f ( 21k , 21i ) : n ≤ k ≤ ∞}, MYi (↓f, n) = max{f ( 21i , 21k ) : n ≤ k ≤ ∞}, M (↓f, n) = max{f (x) : x ∈ Tn,n }. It is easy to verify that M , MXi and MYi are continuous for any i ∈ N.

Lemma 7. For each n ∈ N, Fn ∈ Z(↓USC(T )) and ↓C(T )\ ↓C0 (T ) ⊂ F .

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Proof. Let ε > 0 be given and n0 ∈ N be fixed. Choose N0 ∈ N such that 2N1 0 < 2ε . We will define a map ↓φ :↓USC(T ) →↓USC(T ) \ Fn0 . For any f ∈ USC(T ), we define φ(f ) on TN0 ,N0 as follows, { φ(f )(t) =

M (↓f, N0 ) t = ( 2N1 0 , 2N1 0 ), 0 t ∈ TN0 ,N0 \ ( 2N1 0 , 2N1 0 );

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and for any 0 ≤ i < N0 , we define φ(f ) on Xi and Yi , respectively by,  1 1 n < N0  f ( 2n , 2i ) 1 1 φ(f )( n , i ) = M (↓f, N0 ) n = N0  Xi 2 2 0 N0 < n ≤ ∞

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 1 1 n < N0  f ( 2i , 2n ) 1 1 φ(f )( i , n ) = M (↓f, N0 ) n = N0  Yi 2 2 0 N0 < n ≤ ∞.

Trivially, d(↓φ, id↓USC(T ) ) < ε. It follows from the continuity of M , MXi and MYi that ↓φ is continuous. Moreover, it is clear that ↓C(T )\ ↓C0 (T ) ⊂ F . ¤ Lemma 8. For each n ∈ N, Gn is closed in ↓USC(T ).

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Proof. We fix n0 ∈ N and assume that there is a consequence (↓fn )n in Gn0 convergent to ↓ f ∈↓ USC(T ) \ Gn0 , then there exists a ∈ T ′ such that f (a) < n10 . Let d0 = n10 − f (a) > 0. Because f ∈ USC(T ), there exists δ > 0 such that if d(x, a) < δ then f (x) < f (a) + d20 . Let ε0 = min{δ, d20 }, then for any n ∈ N, it is evident that there not exists a point P ∈↓f such that d(P, (a, fn (a))) < ε0 . This contradicts with the assumption that (↓fn )n convergent to ↓f . We are done. ¤ Lemma 9. For each n ∈ N, Gn ∈ Z(↓USC(T )) ∩ Z(Gn+1 ).

Proof. For each n ∈ N, it directly follows from Gn ⊂ Fn , Lemmata 7 and 8 that 1 Gn ∈ Z(↓USC(T )). Replacing 0 in the three formulas in Proof of Lemma 7 by n+1 , ¤ we may show that Gn ∈ Z(Gn+1 ). Lemma 10. For each n ∈ N, Gn is an AR. Proof. The function ↓r :↓USC(T ) → Gn defined by { max{f (x), n1 } x ∈ T ′ r(f )(x) = f (x) others is a retraction from ↓USC(T ) to Gn . Hence Gn is an AR. Lemma 11. For each n ∈ N, Gn ∼ = Q.

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Proof. For two given continuous functions f and g : Ik → Gn and any ε > 0, choose ′ ′ N ∈ N such that 21N < ε. Define f and g : Ik → Gn as follows,  1 1 1  max{ε, f (q)(1, 2N ), f (q)(1, 2N +1 )} x = (1, 2N ) ′ f (q)(x) = 0 x = (1, 2N1+1 )  f (q)(x) others  1 1 1  max{ε, f (q)(1, 2N ), f (q)(1, 2N +1 )} x = (1, 2N +1 ) ′ g (q)(x) = 0 x = (1, 21N )  f (q)(x) others. ′

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Then it is clear that dH (↓f, ↓f ) < ε, dH (↓g, ↓g ) < ε and f (Ik ) ∩ g (Ik ) = ∅. Hence, Gn has the disjoint-cells property. Therefore, by Lemmata 8, 10 and 2 we conclude that Gn ∼ ¤ = Q. Lemma 12. (Gn )n is a tower with the deformation property in ↓USC(T ).

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Proof. For any ↓f ∈↓USC(T ) and t ∈ I, let H(↓f, t) =↓f ∪ (T × [0, t]), then it is easy to see that H is as required. ¤ The proof of Theorem 1. It follows from the above lemmata. 4. Proof of Theorem 2

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A space is called an absolute Fσδ -space if it is an Fσδ -set in any space which contains it as a subspace. We use Fσδ to denote the class of all absolute Fσδ -spaces, then c0 is an Fσδ absorber of Q [7]. ↓C>0 (T ) =↓C(T )\ ↓C0 (T ) is contained in a σZ-set of ↓USC(T ) by Lemma 7. Furthermore, ↓C>0 (T ) ∈ Fσδ , since ↓C>0 (T ) is an Fσ subset of ↓C(T ) and ↓C(T ) ∈ Fσδ [12]. Therefore, according to Definition 6, it suffices to show that (↓USC(T ), ↓C>0 (T ))is strongly (M0 , Fσδ )-universal. Firstly, we give a lemma in [13]. Lemma 13 ([13]). Let Qu = [0, 1]ω and c1 = {(xn ) ∈ Qu : lim xn = 1}. Then the

pair (Qu , c1 ) is strongly (M0 , Fσδ )-universal.

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Lemma 14. (↓USC(T ), ↓C>0 (T )) is strongly (M0 , Fσδ )-universal.

Proof. Let(C, A) ∈ (M0 , Fσδ ), and ↓ F : C →↓ USC(T) be a continuous map, 0 < ε < 1 and K be a compact subset of C such that ↓F |K : K →↓USC(T) is a Z-embedding. Without loss of generality, we assume that ↓F (C \ K)∩ ↓F (K) = ∅ [3, Lemma 1.1]. By Lemma 13, there exists a continuous function g : C → Qu such that g is a Z-embedding with g −1 (c1 ) = A. Define a function δ : C → [0, 1), by δ(c) = min{ 2ε , 12 dH (↓F (c), ↓F (K))}. It is clear that δ is continuous and δ(c) = 0 if and only if c ∈ K.

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1 For each c ∈ Ck = {c ∈ C : 21k ≤ δ(c) ≤ 2k−1 }, k = 1, 2, . . ., we define the function Hk (c) ∈ USC(T ) by the following formulas. Let t = 2 − 2k δ(c), m = 2−1 n − k − 2.  n ≤ 2k F (c)( 21n , 1)    1   (1 − t)MX1 (↓F (c), n − 2) + F (c)( 2n , 1)t n = 2k + 1,     (1 − t)δ(c)g(c)(1) + F (c)( 21n , 1)t n = 2k + 2,     n = 2k + 3,  MX1 (↓F (c), n − 2) 1 Hk (c)( n , 1) = δ(c)g(c)(1) n = 2k + 4,  2   (1 − t)δ(c) + MX1 (↓F (c), n − 2)t n = 2k + 5,     δ(c)[(1 − t)g(c)(m + 1) + g(c)(m)t] n ≥ 2k + 6, and     n is even,    δ(c) others,   F (c)( 21n , 21n ) n ≤ 2k   1 1   (1 − t)M (↓F (c), n − 2) + F (c)( 2n , 2n )t n = 2k + 1,    1 1  n = 2k + 2,  F (c)( 2n , 2n )t 1 1 Hk (c)( n , n ) = M (↓F (c), n − 2) n = 2k + 3,  2 2   0 n = 2k + 4,     M (↓F (c), n − 2)t n = 2k + 5,    0 n ≥ 2k + 6,   F (c)( 21i , 21n ) 0 < n < i ≤ 2k,   1 1   (1 − t)MXn (↓F (c), i − 2) + F (c)( 2i , 2n )t 0 < n < i = 2k + 1,    1 1  0 < n < i = 2k + 2,  F (c)( 2i , 2n )t 1 1 Hk (c)( i , n ) = MXn (↓F (c), i − 2) 0 < n < i = 2k + 3,  2 2   0 0 < n < i = 2k + 4,     MXn (↓F (c), i − 2)t 0 < n < i = 2k + 5,    0 0 < n < i ≥ 2k + 6,   F (c)( 21n , 21i ) n < i ≤ 2k,   1 1   (1 − t)MYn (↓F (c), i − 2) + F (c)( 2n , 2i )t n < i = 2k + 1,    1 1  n < i = 2k + 2,  F (c)( 2n , 2i )t 1 1 Hk (c)( n , i ) = M (↓F (c), i − 2) n < i = 2k + 3,  Yn 2 2   0 n < i = 2k + 4,     MYn (↓F (c), i − 2)t n < i = 2k + 5,    0 n < i ≥ 2k + 6. ∩ 1 If there exists k ∈ N such that c ∈ Ck Ck−1 , then δ(c) = 2k−1 , and it is easy to verify that Hk (c) = Hk−1 (c). Hence, we can define a map ↓H : C →↓USC(T) as follow. If c ∈ Ck , let ↓H(c) =↓Hk (c); if c ∈ K, let ↓H(c) =↓F (c). To show ↓H is as required, we need the following claims: Claim 1. ↓H −1 [↓C>0 (T )] \ K = A \ K. It is clear that c ∈ A if and only if lim g(c)(m) = 1 if and only if lim δ(c)[(1 − m→∞

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t)g(c)(m + 1) + tg(c)(m)] = δ(c). Moreover, if c ∈ A \ K, then H(c)(0, 1) =


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δ(c) > 0, and H(c) is continuous at the point (0, 1) by lim δ(c)[(1 − t)g(c)(m + m→∞

1) + tg(c)(m)] = δ(c). It is clear that H(c) is continuous at other points. Hence, c ∈↓H −1 [↓C>0 (T )] \ K. On the other hand, if c ∈↓H −1 [↓C>0 (T )] \ K, then H(c) ∈ C>0 (T ), H(c) is continuous at the point (0, 1), lim δ(c)[(1 − t)g(c)(m + 1) + m→∞

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tg(c)(m)] = δ(c). Hence c ∈ A. Claim 2. For any c ∈ C, dH (↓H(c), ↓F (c)) ≤ δ(c) < ε. If c ∈ K, it is trivial. If c ∈ C \ K, then there exists k, such that c ∈ Ck and 1 ) = H(c) = Hk (c). H(c)( 21i , 21j ) = F (c)( 21i , 21j ) for any i, j ≤ 2k. H(c)( 21i , 22k+3 1 1 1 1 1 1 max{F (c)( 2i , 2n ) : 2k < n ≤ ∞} and H(c)( 22k+3 , 2i ) = max{F (c)( 2n , 2i ) : 2k < 1 1 n ≤ ∞} for any i ≤ 2k. H(c)( 22k+3 , 22k+3 ) = max{F (c)( 21i , 21j ) : 2k < i, j ≤ ∞}. For any point p = ((a, b), y) ∈↓F (c), it is evident that there is a point 1 1 1 1 1 1 q ∈ {((a, b), y), (( i , 2k+3 ), y), (( 2k+3 , i ), y), (( 2k+3 , 2k+3 ), y)}∩ ↓H(c) 2 2 2 2 2 2 1 1 such that d(p, q) ≤ 22k+1 < 2k ≤ δ(c), that is, B(↓H(c), δ(c)) ⊃↓F (c). On the other hand, max{H(c)( 21j , 21n ) : 2k < n ≤ ∞} ≤ max{F (c)( 21j , 21n ) : 2k < n ≤ ∞} and max{H(c)( 21n , 21j ) : 2k < n ≤ ∞} ≤ max{max{F (c)( 21n , 21j ) : 2k < n ≤ ∞}, δ(c)} for any j ≤ 2k. max{H(c)( 21i , 21j ) : 2k < i, j ≤ ∞} ≤ max{F (c)( 21i , 21j ) : 2k < i, j ≤ ∞}. By a similar discussion as above, we have B(↓ F (c), δ(c)) ⊃↓ H(c). Therefore, dH (↓H(c), ↓F (c)) ≤ δ(c) < ε for all c ∈ C. Claim 3. ↓H : C →↓USC(T ) is continuous. If c ∈ Ck , it follows from the continuity of ↓F, M, MXi , MYi , g and δ that ↓H is continuous at point c. If c ∈ K, ↓H is continuous at c, since ↓F is continuous ′ ′ ′ ′ ′ and dH (↓F (c ), ↓H(c )) ≤ δ(c ) ≤ 21 dH (↓F (c ), ↓F (K)) for all c ∈ C. Hence ↓H is continuous. Claim 4. ↓H : C →↓USC(T ) is a injection. Let c1 , c2 ∈ C and c1 ̸= c2 . We shall prove ↓H(c1 ) ̸=↓H(c2 ). Since ↓H|K =↓F |K is an injection, it suffices to consider the following two cases: case 1. c1 ∈ K, c2 ∈ C \ K. By claim 2, we have dH (↓ F (c2 ), ↓ H(c2 )) ≤ δ(c2 ) ≤ 21 dH (↓ F (c2 ), ↓ F (K)) ≤ 1 d (↓ F (c2 ), ↓ F (c1 )) = 21 dH (↓ F (c2 ), ↓ H(c1 )). Thus, dH (↓ H(c1 ), ↓ H(c2 )) ≥ 2 H dH (↓ F (c2 ), ↓ H(c1 )) − dH (↓ F (c2 ), ↓ H(c2 )) ≥ 21 dH (↓ F (c2 ), ↓ H(c1 )) ≥ δ(c2 ) > 0. We are done. case 2. c1 , c2 ∈ C \ K. Assume ↓ H(c1 ) =↓ H(c2 ), then δ(c1 ) = H(c1 )(0, 1) = H(c2 )(0, 1) = δ(c2 ), therefore, there exists k ∈ N such that c1 , c2 ∈ Ck , 21k ≤ δ(c1 ) = δ(c2 ) ≤ 1 . We shall prove that g(c1 )(m) = g(c2 )(m) for any m ∈ N by induction. 2k−1 1 1 Firstly, since Hk (c1 )( 22k+4 , 1) = Hk (c2 )( 22k+4 , 1) and δ(c1 ) = δ(c2 ) > 0, we conclude that g(c1 )(1) = g(c2 )(1). Assume that g(c1 )(m) = g(c2 )(m). Then by 1 1 , 1) = Hk (c2 )( 22m+2k+4 , 1), we have Hk (c1 )( 22m+2k+4 δ(c1 )[(1 − t)g(c1 )(m + 1) + g(c1 )(m)t] = δ(c2 )[(1 − t)g(c2 )(m + 1) + g(c2 )(m)t].

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If t ̸= 1, then by assumption g(c1 )(m + 1)) = g(c2 )(m + 1). If t = 1, then 1 1 δ(c1 )g(c1 )(m + 1) = Hk (c1 )( 22m+2k+6 , 1) = Hk (c2 )( 22m+2k+6 , 1) = δ(c2 )g(c2 )(m + 1), that is, g(c1 (m + 1)) = g(c2 )(m + 1). Therefore, g(c1 )(m) = g(c2 )(m) for any m ∈ N. Hence g(c1 ) = g(c2 ). Thus c1 = c2 since g is injection. It contradicts c1 ̸= c2 . Claim 5. ↓H is a Z-embedding. For any c ∈ C \ K, by the definition of H, there exists n ∈ N such that H(c)( 21n , 21n ) = 0. Since ↓ H(K) is a Z-set of ↓ USC(T ), ↓ H(C) is also a Z-set of ↓USC(T ) (cf. [13, Lemma 5]). ¤

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The Proof of Theorem 2. It follows from Lemmata 1 and 14.

References

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[1] R.D. Anderson, On sigma-compact subsets of infinite-dimensional manifolds, unpublished manuscript. [2] J. Baars, H. Gladdines and J. van Mill, Absorbing systems in infinite-dimensional manifold, Topology Appl. 50 (1993), 147-182. [3] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional absoulte retracts, Michigan Math. J. 33 (1986), 291-313. [4] T. Banakh, T. Radul, M. Zarichnyi, Absorbing sets in infinite-dimensional manifolds, VNTL Publishers, Lviv, 72 (1976), 515-519. [5] D.W. Curtis, Boundary sets in the Hilbert cube, Topology Appl. 20 (1985), 201-221. [6] D.W. Curtis and R.M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38. [7] J.J. Dijkstra, J. van Mill and J. Mogilski, The space of infinite-dimensional compacta and other topological copys of (lf2 )w , Pacific J. Math. 152 (1992), 255-273. [8] A. Illanes and S.B. Nadler, Jr., Hyperspaces, Fundamental and Recent Advances, Pure and Applied Mat. 216, Marcel Dekker, Inc., New York, 1999. [9] J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library 43, Elsevier Sci. Publ. B.V., Amsterdam, 1989. [10] J. van Mill, The Infinite-Dimensional Topology of Function Spaces, Pre-requisites and Introduction, North-Holland Math. Library 43, Elsevier Sci. Publ. B.V. , Amsterdam, 1989. [11] H. Toru´ nczyk, On CE-images of the Hilbert cube and characterizations of Q-manifolds, Fund. Math. 106 (1980), 431–437. [12] Z. Yang, The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification, Science in China, Ser. A: Mathematics 48 (2005), 468-484. [13] Z. Yang, The hyperspace of the regions below of continuous maps is homeomorphic to c0 , Topology Appl. 153 (2006), 2908–2921. [14] Z. Yang and X. Zhou, (↓USC(X), ↓C(X)) ∼ = (Q, c0 ) if and only if X is a compactum without dense set of isolated points, Topology Appl. (2007), doi:10.1016/j.topol.2006.12.013. [15] Z. Yang and L. Fan, The hyperspace of the regions below of continuous maps from the converging sequence, Northeast Math. J. 22(1), 45-54.

The Hyperspace of the regions below of continuous maps from S × S to I 11

(Zhongqiang Yang) Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China P.R. E-mail address: [email protected]

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(Nada Wu) Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China P.R. E-mail address: [email protected]

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Received December 23, 2005 and revised September 27, 2007