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Continuous selections for proximal continuous paraconvex-valued mappings
Topology and its Applications 169 (2014) 45–50
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Continuous selections for proximal continuous paraconvex-valued mappings Takamitsu Yamauchi Department of Mathematics, Shimane University, Matsue, 690-8504, Japan
a r t i c l e
i n f o
MSC: primary 54C65 secondary 54C60 Keywords: Set-valued mapping Continuous selection Proximal continuous Paraconvex
a b s t r a c t We show that every metric-proximal continuous paraconvex- and closed-valued mapping from an arbitrary topological space into a Banach space admits a continuous selection, which answers a question of Gutev. To show the result, we apply the selection factorization property introduced by Nedev. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Throughout this paper, let X denote a topological space, (Y, ρ) a metric space, 2Y the set of all non-empty subsets of Y and F (Y ) the set of all non-empty closed subsets of Y . If Y is a normed space, ρ is assumed to be the metric induced by its norm. For a mapping Φ : X → 2Y , a mapping f : X → Y is called a selection for Φ if f (x) ∈ Φ(x) for each x ∈ X. As a generalization of his convex-valued selection theorem [7, Theorem 3.2 ], Michael [8, Theorem 2.1] established a selection theorem for paraconvex-valued mappings on paracompact spaces. Here, for α > 0, a closed subset P of a normed space Y is said to be α-paraconvex if, whenever p ∈ Y and r > 0 satisfy ρ(p, P ) < r, ρ(q, P ) αr
for all q ∈ conv B(p, r) ∩ P ,
where ρ(q, P ) = inf{ρ(q, y): y ∈ P }, B(p, r) = {y ∈ Y : ρ(p, y) < r} and conv(A) is the convex hull of A. A closed subset is 0-paraconvex if and only if it is convex. Klee [5] proved that every closed subset of a normed space Y is 1-paraconvex if and only if Y is either an inner product space or two-dimensional. The set P is said to be paraconvex if it is α-paraconvex for some α < 1. For example, the letters V, X, Y and Z and a circular arc subtending an angle < π are paraconvex, while the letter U and a circular arc subtending an angle π are not paraconvex (see [8, Examples 1.1 and 1.2]). E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2014.02.031 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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For a normed space Y and α 0, let Fα (Y ) denote the set of all non-empty closed α-paraconvex subsets of Y . A mapping Φ : X → 2Y is said to be lower semicontinuous (l.s.c. for short) if for every open subset V of Y , the set Φ−1 [V ] = {x ∈ X: Φ(x) ∩ V = ∅} is open in X. For a mapping Φ : X → 2Y and r > 0, a mapping f : X → Y is called an r-selection for Φ if ρ(f (x), Φ(x)) < r for each x ∈ X. Michael established the following theorem. Theorem 1.1. (Michael [8, Theorem 2.1]) Let X be a paracompact space, Y a Banach space, α ∈ [0, 1) and Φ : X → Fα (Y ) an l.s.c. mapping. Then: (a) There exists a continuous selection for Φ. (b) If there exists a continuous r-selection g : X → Y for Φ for some r > 0, then there exists a continuous ∞ selection f : X → Y for Φ such that ρ(g(x), f (x)) < α ˆ r for all x ∈ X, where α ˆ = 1 + i=0 αi . For paraconvex sets and related selection theorems, see [4,6] and [12–17]. In [6], Makala proved a collectionwise normal version of Theorem 1.1, which is also a generalization of [7, Theorem 3.2 ]. He also gave some generalizations for Theorem 1.1. In this context, Gutev posed a question concerning paraconvex-valued ρ-proximal continuous mappings. Here, a mapping Φ : X → 2Y is said to be ρ-upper semicontinuous (ρ-u.s.c. for short) if for every x ∈ X and ε > 0, there exists a neighborhood U of x such that Φ(z) ⊂ B(Φ(x), ε) for each z ∈ U . A mapping Φ is said to be ρ-proximal continuous if it is l.s.c. and ρ-u.s.c. It should be remarked that Gutev [2, Theorem 6.1] proved that for every topological space X and every Banach space Y , if a mapping Φ : X → F0 (Y ) is d-proximal continuous for some metric d on Y compatible with its topology, then Φ admits a continuous selection. In [6, Question 3.7], Gutev asked the following question. Question 1 (Gutev). Let X be a topological space, Y a Banach space, and Φ : X → Fα (Y ) a ρ-proximal continuous mapping for some α ∈ [0, 1). Then, is it true that Φ admits a continuous selection? The main purpose of this paper is to give an affirmative answer for Question 1 by proving the following theorem. Theorem 1.2. Let X be a topological space, Y be a Banach space, α ∈ [0, 1) and Φ : X → Fα (Y ) a ρ-proximal mapping. Then: (a) There exists a continuous selection for Φ. (b) If there exists a continuous r-selection g : X → Y for Φ for some r > 0, then there exists a continuous ∞ selection f : X → Y for Φ such that ρ(g(x), f (x)) < α ˆ r for all x ∈ X, where α ˆ = 1 + i=0 αi . The author does not know whether “ρ-proximal continuous” in Theorem 1.2 can be replaced by “d-proximal continuous for some metric d on Y compatible with its topology” as in Gutev’s theorem [2, Theorem 6.1]. Proof of Theorem 1.2 is given in the next section. In Section 3, we show some results on ρ-proximal continuous mappings. Let R denote the space of all real numbers with the usual topology, and N the set of all positive integers. The diameter of a subset A of a metric space is denoted by diam A, and for a collection U of a subsets of a metric space, let mesh U = sup{diam U : U ∈ U }. For undefined notation and terminology, we refer to [1].
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2. Proof of Theorem 1.2 To prove our main theorem, we apply the selection factorization property introduced by Nedev [11]. Nedev defined the property for normal spaces using locally finite open covers. We use cozero-set (functionally open set) covers to apply the property for general topological spaces. For a topological space X and a metrizable space Y , we say that a mapping Φ : X → 2Y satisfies the functional selection factorization property (f.s.f.p. for short) if for every closed subset F of X and every locally finite collection V of open subsets of Y with F ⊂ Φ−1 [ V ], there exists a locally finite cozero-set (in F ) cover U of F which refines the collection {Φ−1 [V ]: V ∈ V }. If X is normal, the f.s.f.p. is equivalent to the selection factorization property due to Nedev [11]. According to [11], we have the following lemma. For a subset A of a normed space Y , let conv(A) denote the closed convex hull of A. Lemma 2.1. Let X be a topological space, Y a Banach space and Φ : X → F (Y ) a mapping with the f.s.f.p. Then the mapping conv Φ : X → F0 (Y ) defined by (conv Φ)(x) = conv(Φ(x)) for x ∈ X admits a continuous selection. Proof. If a cover {Uα : α ∈ A} of space is refined by a locally finite cozero-set cover, then there exists a locally finite cozero-set cover {Vα : α ∈ A} such that Vα ⊂ Uα for each α ∈ A (see [3, Theorem 1.4]). Thus the conclusion is obtained by repeating the proof of [11, Lemma 3.1 and Proposition 4.3]. 2 Applying [2, Theorem 3.1], we have the following. Lemma 2.2. Let X be a topological space and (Y, ρ) a metric space. Then every ρ-proximal continuous mapping Φ : X → 2Y satisfies the f.s.f.p. Proof. Let F be a closed subset of X and V a locally finite collection of open subsets of Y such that F ⊂ Φ−1 [ V ]. By virtue of [2, Theorem 3.1], there exists a σ-discrete (in X) collection W of cozero-sets of X such that W refines {Φ−1 [V ]: V ∈ V } and W = Φ−1 [ V ]. According to [9, Theorem 1.2] and [10, Theorem 1.2] (see also [3, Theorem 1.2]), the σ-discrete cozero-set cover {W ∩ F : W ∈ W } of F is refined by a locally finite cozero (in F ) cover U of F . Then U is the required cover. 2 The following lemma is easy to verify and we omit the proof. Lemma 2.3. Let X be a topological space, (Y, ρ) a metric space, Φ : X → F (Y ) a ρ-proximal continuous mapping and g : X → Y a continuous mapping. Then the mapping u : X → R defined by u(x) = ρ(g(x), Φ(x)) for x ∈ X is continuous. For Φ : X → 2Y , A ⊂ X and a selection h : A → Y for Φ|A , define Φh : X → 2Y by Φh (x) = {h(x)} if x ∈ A and Φh (x) = Φ(x) otherwise. The following technical theorem is a key to show Theorem 1.2. Theorem 2.4. Let X be a topological space, (Y, ρ) a metric space, Φ : X → 2Y a ρ-proximal continuous mapping, C a cozero-set of X, h : C → Y a continuous selection for Φ|C and Z a zero-set of X with Z ⊂ C. Also, let r > 0 and let g : X → Y be a continuous r-selection for Φh|Z . Then the mapping ϕ : X → 2Y defined by ϕ(x) = Φh|Z (x) ∩ B(g(x), r) for x ∈ X satisfies the f.s.f.p. Proof. Let F be a closed subset of X and V a locally finite open collection in Y such that F ⊂ −1 {ϕ [V ]: V ∈ V }. Since C is a cozero-set of X and ϕ(x) = {h(x)} for each x ∈ Z, h−1 ( V ) is a cozero set of X such that F ∩ Z ⊂ h−1 ( V ). Choose a cozero-set O of F such that F ∩ Z ⊂ O ⊂ O ⊂ h−1 ( V ) and put U0 = {h−1 (V ) ∩ O: V ∈ V }. Then U0 is a locally finite cozero-set collection of F such that U0
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refines {Φ−1 ∈ V } and F ∩ Z ⊂ U0 . Since F \ U0 is a zero-set of F and F \ Z is a cozero-set h|Z [V ]: V of F such that F \ U0 ⊂ F \ Z, there exist a cozero-set O of F and a closed subset E of F such that F \ U0 ⊂ O ⊂ E ⊂ F \ Z. We shall take a locally finite cozero-set (in E) cover G of E which refines {Φ−1 [V ]: V ∈ V }. Put U1 = {G ∩ O : G ∈ G }. Then U0 ∪ U1 is the required locally finite cozero-set (in F ) collection. Here, we show the existence of such a cover G of E. Note that Φh|Z (x) = Φ(x) for each x ∈ E since E ∩ Z = ∅. For each n ∈ N, put Un = x ∈ X: ρ g(x), Φ(x) < r − 1/n .
(2.1)
By Lemma 2.3, the collection {Un : n ∈ N} is a cozero-set cover of X. For each n ∈ N, take a locally finite open cover Wn of Y such that mesh Wn < 1/n. For each n ∈ N and W ∈ Wn , put n OW =
Φ−1 V ∩ B(W, r − 1/n) : V ∈ V
and
n GnW = Un ∩ g −1 (W ) ∩ OW .
(2.2) (2.3)
n n Since OW = Φ−1 [( V ) ∩ B(W, r − 1/n)], according to [2, Lemma 3.3], OW is a cozero-set of X. Thus, −1 n since {g (W ): W ∈ Wn } is a locally finite cozero-set cover of X, so is {GW : W ∈ Wn } for each n ∈ N. We claim that the collection {GnW : W ∈ Wn , n ∈ N} covers E. For, let x ∈ E. Then x ∈ ϕ−1 [V ] for some V ∈ V . Take y ∈ ϕ(x) ∩ V = Φ(x) ∩ B(g(x), r) ∩ V . Then ρ(g(x), y) < r. Choose n ∈ N satisfying ρ(g(x), y) < r − 1/n. Since Wn covers Y , we may take W ∈ Wn such that g(x) ∈ W . Then n . Since x ∈ g −1 (W ). Since y ∈ Φ(x) ∩ V ∩ B(W, r − 1/n), we have x ∈ Φ−1 [( V ) ∩ B(W, r − 1/n)] = OW n ρ(g(x), Φ(x)) ρ(g(x), y) < r − 1/n, we have x ∈ Un . Thus x ∈ GW . Fix n ∈ N and W ∈ Wn . We claim that there exists a σ-locally finite cozero-set (in E) collection UWn of E such that E ∩ GnW ⊂ UWn and UWn refines {Φ−1 [V ∩ B(W, r − 1/n)]: V ∈ V }. Indeed, since GnW is a cozero-set of X, we may take a countable collection {Ck : k ∈ N} of cozero-sets of E such that E ∩ GnW = k∈N Ck = k∈N Ck . For each k ∈ N, n E ∩ Ck ⊂ OW =
Φ−1 V ∩ B(W, r − 1/n) : V ∈ V
by (2.2) and (2.3). Since Φ satisfies the f.s.f.p. by Lemma 2.2, there exists a locally finite cozero-set (in E ∩ Ck ) cover Ck of E ∩ Ck such that Ck refines {Φ−1 [V ∩ B(W, r − 1/n)]: V ∈ V }. Then UWn = {C ∩ Ck : C ∈ Ck , k ∈ N} is the required σ-locally finite cozero-set (in E) collection. Put U = {U ∩ GnW : U ∈ UWn , W ∈ Wn , n ∈ N}. Then U is a σ-locally finite cozero-set (in E) cover of E. By virtue of [9, Theorem 1.2] and [10, Theorem 1.2] (see also [3, Theorem 1.2]), U is refined by a locally finite cozero-set (in E) cover G of E. To see that G is the required cover, it suffices to show that U refines {ϕ−1 [V ]: V ∈ V }. Let U ∈ U . Then U = U ∩ GnW for some U ∈ UWn , W ∈ Wn and n ∈ N. By the choice of UWn , we have U ⊂ Φ−1 [V ∩ B(W, r − 1/n)] for some V ∈ V . To see that U ⊂ ϕ−1 [V ], let x ∈ U = U ∩ GnW . By (2.1) and (2.3), we have ρ(g(x), Φ(x)) < r − 1/n and g(x) ∈ W . Also, we may take z ∈ Φ(x) ∩ V ∩ B(W, r − 1/n) because x ∈ U ⊂ Φ−1 [V ∩ B(W, r − 1/n)]. Choose y ∈ W such that ρ(y, z) < r − 1/n. Since mesh Wn < 1/n, we have ρ(g(x), y) < 1/n, and hence ρ(g(x), z) < r. Thus z ∈ Φ(x) ∩ B(g(x), r) ∩ V = ϕ(x) ∩ V , which implies x ∈ ϕ−1 [V ], and hence U ⊂ ϕ−1 [V ]. Therefore, ϕ satisfies the f.s.f.p. 2 Now, we prove Theorem 1.2. Proof of Theorem 1.2. By repeating the proof of (b) in [8, Theorem 2.1] with applying Lemma 2.1 and Theorem 2.4, we have the following:
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(∗) Let X be a topological space, Y a Banach space, α ∈ [0, 1), Φ : X → Fα (Y ) a ρ-proximal mapping, C a cozero-set of X, h : C → Y a continuous selection for Φ|C and Z a zero-set of X with Z ⊂ C. If there exists a continuous r-selection g : X → Y for Φh|Z for some r > 0, then there exists a continuous ∞ ˆ for all x ∈ X, where α ˆ = 1 + i=0 αi . selection f : X → Y for Φh|Z such that ρ(g(x), f (x)) < αr The statement (b) follows from (∗) with C = ∅. For (a), choose λ 2 such that ρ(0, Φ(x)) < λ for some x ∈ X, and let β = max{ˆ α, λ}. For each n ∈ N, put An = x ∈ X: ρ 0, Φ(x) β n−1 Bn = x ∈ X: ρ 0, Φ(x) < β n .
and
By Lemma 2.3, An is a zero-set and Bn is a cozero-set of X. Moreover, An ⊂ Bn ⊂ An+1 for n ∈ N and X = n∈N Bn . It suffices to construct continuous selections fn : Bn+1 → Y , n ∈ N, for Φ|Bn+1 such that fn+1 |Bn = fn |Bn for each n ∈ N. Then the mapping f : X → Y defined by f |Bn = fn |Bn for n ∈ N is a selection for Φ. Since {Bn : n ∈ N} is an open cover of X, f is continuous (see [6, Remark 2.4]). As in the proof of (a) in [8, Theorem 2.1], we construct the mappings fn , n ∈ N, by induction on n. To keep the induction going, we shall also require ρ(fn (x), 0) < β n+2 for every n ∈ N and x ∈ Bn+1 . The existence of f1 : B2 → Y follows from (b), with X replaced by B2 , r by β 2 , and with g(x) = 0 for every x ∈ A2 . Then ρ(f1 (x), 0) < α ˆ β 2 β 3 for every x ∈ B2 . Assume that fn : Bn+1 → Y has been constructed. n+2 for each x ∈ Bn+2 . By applying (∗) with replacing X by Bn+2 , C by Bn+1 , Then ρ(0, Φfn |An+1 (x)) < β n+2 h by fn , Z by An+1 , r by β , and with g(x) = 0 for every x ∈ Bn+2 , we have a continuous selection ˆ β n+2 β n+3 . Then fn+1 is also a selection fn+1 : Bn+2 → Y for Φfn |An+1 |Bn+2 such that ρ(fn+1 (x), 0) < α for Φ|Bn+2 . Since fn+1 |An+1 = fn |An+1 , we have fn+1 |Bn = fn |Bn . Thus fn+1 is the required mapping. 2 Remark 2.5. Semenov [16, Theorem 6] generalized Michael’s paraconvex-valued selection theorem (Theorem 1.1). Theorem 1.2 can be generalized similarly. For a non-empty closed subset P of a Banach space Y , r > 0 and y ∈ Y , we put δ P, B(y, r) = sup ρ(q, P )/r: q ∈ conv P ∩ B(y, r) , where we let δ(P, B(y, r)) = 0 if P ∩ B(y, r) = ∅. The function αP : (0, ∞) → [0, ∞) defined by αP (r) = sup δ P, B(y, r) : y ∈ Y is called the function of nonconvexity of P . Note that a closed subset P of Y is α-paraconvex if and only if αP (r) α for each r ∈ (0, ∞). A function β : (0, ∞) → [0, ∞) is said to be geometrically summable if for ∞ each t ∈ (0, ∞), the series n=0 qβn (t) converges, where the sequence {qβn (t)} is defined as follows: qβ0 (t) = t,
qβn (t) = β qβn−1 (t) · qβn−1 (t),
n ∈ N.
By an argument analogous to the proof of [16, Theorem 6] with applying Lemma 2.1 and Theorem 2.4, “Φ : X → Fα (Y )” in Theorem 1.2 can be relaxed to “Φ : X → F (Y ) with some geometrically summable function β : (0, ∞) → [0, ∞) such that αΦ(x) (t) < β(t) for each x ∈ X and t ∈ (0, ∞)”. 3. Other results on ρ-proximal continuous mappings The following corollary is immediate from Theorem 2.4.
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Corollary 3.1. Let X be a topological space, (Y, ρ) a metric space, Φ : X → 2Y a ρ-proximal continuous mapping, r > 0 and g : X → Y a continuous r-selection for Φ. Then the mapping ϕ : X → 2Y defined by ϕ(x) = Φ(x) ∩ B(g(x), r) for x ∈ X satisfies the f.s.f.p. Thus, by Lemma 2.1, we have the following. Corollary 3.2. Let X be a topological space, Y a Banach space and Φ : X → F0 (Y ) a ρ-proximal continuous mapping, r > 0 and g : X → Y a continuous r-selection for Φ. Then Φ admits a continuous selection f : X → Y such that ρ(g(x), f (x)) r for each x ∈ X. A mapping Ψ is said to be upper semicontinuous (u.s.c. for short) if for every closed subset F of Y , Φ−1 [F ] is closed in X. It is easy to see that every u.s.c. mapping is ρ-u.s.c. As in Lemma 2.1, by repeating the proof of [11, Lemma 3.1 and Proposition 4.1], we have the following lemma. Let C (Y ) denote the set of all non-empty compact subsets of Y . Lemma 3.3. Let X be a topological space, Y a completely metrizable space, Φ : X → F (Y ) a mapping with the f.s.f.p. Then there exist a u.s.c. mapping ψ : X → C (Y ) and an l.s.c. mapping ϕ : X → C (Y ) such that ϕ(x) ⊂ ψ(x) ⊂ Φ(x) for each x ∈ X. For a mapping Φ : X → 2Y , a mapping ϕ : X → 2Y is called a multi-selection for Φ if ϕ(x) ⊂ Φ(x) for each x ∈ X. By Corollary 3.1 and Lemma 3.3, we have the following corollary which answers a question asked by Gutev to the author. Corollary 3.4. Let X be a topological space, Y a completely metrizable space, Φ : X → F (Y ) a ρ-proximal continuous mapping, g : X → Y a continuous mapping and r > 0 such that ρ(g(x), Φ(x)) < r for all x ∈ X. Then there exist a u.s.c. multi-selection ψ : X → C (Y ) for Φ and an l.s.c. mapping ϕ : X → C (Y ) such that ϕ(x) ⊂ ψ(x) ⊂ B(g(x), r) for each x ∈ X. Acknowledgements The author would like to thank the referee for valuable comments and suggestions. References [1] R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. [2] V. Gutev, Weak factorizations of continuous set-valued mappings, Topol. Appl. 102 (2000) 33–51. [3] T. Hoshina, Extensions of mappings II, in: K. Morita, J. Nagata (Eds.), Topics in General Topology, North-Holland, Amsterdam, 1989, pp. 41–80. [4] G.E. Ivanov, Continuous selections of multifunctions with weakly convex values, Topol. Appl. 155 (2008) 851–857. [5] V. Klee, Circumspheres and inner products, Math. Scand. 8 (1960) 363–370. [6] N.R.L. Makala, Selections for paraconvex-valued mappings on non-paracompact domains, Topol. Appl. 159 (2012) 153–157. [7] E. Michael, Continuous selections I, Ann. Math. 63 (1956) 361–382. [8] E. Michael, Paraconvex sets, Math. Scand. 7 (1959) 372–376. [9] K. Morita, Paracompactness and product spaces, Fundam. Math. 50 (1962) 223–236. [10] K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964) 365–382. [11] S. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980) 291–317. [12] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht, 1998. [13] D. Repovš, P.V. Semenov, Continuous selections as uniform limits of δ-continuous ε-selections, Set-Valued Anal. 7 (1999) 239–254. [14] D. Repovš, P.V. Semenov, On continuous choice of retractions onto nonconvex subsets, Topol. Appl. 157 (2010) 1510–1517. [15] P.V. Semenov, Functionally paraconvex sets, Math. Notes 54 (1993) 1236–1240. [16] P.V. Semenov, Fixed-point theorems with controllable rejection of the convexity of the values of a multivalued mapping, Sb. Math. 189 (1998) 461–480. [17] P.V. Semenov, Nonconvexity in problems of multivalued calculus, J. Math. Sci. (N.Y.) 100 (2000) 2682–2699.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Continuous selections for proximal continuous paraconvex-valued mappings Takamitsu Yamauchi Department of Mathematics, Shimane University, Matsue, 690-8504, Japan
a r t i c l e
i n f o
MSC: primary 54C65 secondary 54C60 Keywords: Set-valued mapping Continuous selection Proximal continuous Paraconvex
a b s t r a c t We show that every metric-proximal continuous paraconvex- and closed-valued mapping from an arbitrary topological space into a Banach space admits a continuous selection, which answers a question of Gutev. To show the result, we apply the selection factorization property introduced by Nedev. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Throughout this paper, let X denote a topological space, (Y, ρ) a metric space, 2Y the set of all non-empty subsets of Y and F (Y ) the set of all non-empty closed subsets of Y . If Y is a normed space, ρ is assumed to be the metric induced by its norm. For a mapping Φ : X → 2Y , a mapping f : X → Y is called a selection for Φ if f (x) ∈ Φ(x) for each x ∈ X. As a generalization of his convex-valued selection theorem [7, Theorem 3.2 ], Michael [8, Theorem 2.1] established a selection theorem for paraconvex-valued mappings on paracompact spaces. Here, for α > 0, a closed subset P of a normed space Y is said to be α-paraconvex if, whenever p ∈ Y and r > 0 satisfy ρ(p, P ) < r, ρ(q, P ) αr
for all q ∈ conv B(p, r) ∩ P ,
where ρ(q, P ) = inf{ρ(q, y): y ∈ P }, B(p, r) = {y ∈ Y : ρ(p, y) < r} and conv(A) is the convex hull of A. A closed subset is 0-paraconvex if and only if it is convex. Klee [5] proved that every closed subset of a normed space Y is 1-paraconvex if and only if Y is either an inner product space or two-dimensional. The set P is said to be paraconvex if it is α-paraconvex for some α < 1. For example, the letters V, X, Y and Z and a circular arc subtending an angle < π are paraconvex, while the letter U and a circular arc subtending an angle π are not paraconvex (see [8, Examples 1.1 and 1.2]). E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2014.02.031 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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For a normed space Y and α 0, let Fα (Y ) denote the set of all non-empty closed α-paraconvex subsets of Y . A mapping Φ : X → 2Y is said to be lower semicontinuous (l.s.c. for short) if for every open subset V of Y , the set Φ−1 [V ] = {x ∈ X: Φ(x) ∩ V = ∅} is open in X. For a mapping Φ : X → 2Y and r > 0, a mapping f : X → Y is called an r-selection for Φ if ρ(f (x), Φ(x)) < r for each x ∈ X. Michael established the following theorem. Theorem 1.1. (Michael [8, Theorem 2.1]) Let X be a paracompact space, Y a Banach space, α ∈ [0, 1) and Φ : X → Fα (Y ) an l.s.c. mapping. Then: (a) There exists a continuous selection for Φ. (b) If there exists a continuous r-selection g : X → Y for Φ for some r > 0, then there exists a continuous ∞ selection f : X → Y for Φ such that ρ(g(x), f (x)) < α ˆ r for all x ∈ X, where α ˆ = 1 + i=0 αi . For paraconvex sets and related selection theorems, see [4,6] and [12–17]. In [6], Makala proved a collectionwise normal version of Theorem 1.1, which is also a generalization of [7, Theorem 3.2 ]. He also gave some generalizations for Theorem 1.1. In this context, Gutev posed a question concerning paraconvex-valued ρ-proximal continuous mappings. Here, a mapping Φ : X → 2Y is said to be ρ-upper semicontinuous (ρ-u.s.c. for short) if for every x ∈ X and ε > 0, there exists a neighborhood U of x such that Φ(z) ⊂ B(Φ(x), ε) for each z ∈ U . A mapping Φ is said to be ρ-proximal continuous if it is l.s.c. and ρ-u.s.c. It should be remarked that Gutev [2, Theorem 6.1] proved that for every topological space X and every Banach space Y , if a mapping Φ : X → F0 (Y ) is d-proximal continuous for some metric d on Y compatible with its topology, then Φ admits a continuous selection. In [6, Question 3.7], Gutev asked the following question. Question 1 (Gutev). Let X be a topological space, Y a Banach space, and Φ : X → Fα (Y ) a ρ-proximal continuous mapping for some α ∈ [0, 1). Then, is it true that Φ admits a continuous selection? The main purpose of this paper is to give an affirmative answer for Question 1 by proving the following theorem. Theorem 1.2. Let X be a topological space, Y be a Banach space, α ∈ [0, 1) and Φ : X → Fα (Y ) a ρ-proximal mapping. Then: (a) There exists a continuous selection for Φ. (b) If there exists a continuous r-selection g : X → Y for Φ for some r > 0, then there exists a continuous ∞ selection f : X → Y for Φ such that ρ(g(x), f (x)) < α ˆ r for all x ∈ X, where α ˆ = 1 + i=0 αi . The author does not know whether “ρ-proximal continuous” in Theorem 1.2 can be replaced by “d-proximal continuous for some metric d on Y compatible with its topology” as in Gutev’s theorem [2, Theorem 6.1]. Proof of Theorem 1.2 is given in the next section. In Section 3, we show some results on ρ-proximal continuous mappings. Let R denote the space of all real numbers with the usual topology, and N the set of all positive integers. The diameter of a subset A of a metric space is denoted by diam A, and for a collection U of a subsets of a metric space, let mesh U = sup{diam U : U ∈ U }. For undefined notation and terminology, we refer to [1].
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2. Proof of Theorem 1.2 To prove our main theorem, we apply the selection factorization property introduced by Nedev [11]. Nedev defined the property for normal spaces using locally finite open covers. We use cozero-set (functionally open set) covers to apply the property for general topological spaces. For a topological space X and a metrizable space Y , we say that a mapping Φ : X → 2Y satisfies the functional selection factorization property (f.s.f.p. for short) if for every closed subset F of X and every locally finite collection V of open subsets of Y with F ⊂ Φ−1 [ V ], there exists a locally finite cozero-set (in F ) cover U of F which refines the collection {Φ−1 [V ]: V ∈ V }. If X is normal, the f.s.f.p. is equivalent to the selection factorization property due to Nedev [11]. According to [11], we have the following lemma. For a subset A of a normed space Y , let conv(A) denote the closed convex hull of A. Lemma 2.1. Let X be a topological space, Y a Banach space and Φ : X → F (Y ) a mapping with the f.s.f.p. Then the mapping conv Φ : X → F0 (Y ) defined by (conv Φ)(x) = conv(Φ(x)) for x ∈ X admits a continuous selection. Proof. If a cover {Uα : α ∈ A} of space is refined by a locally finite cozero-set cover, then there exists a locally finite cozero-set cover {Vα : α ∈ A} such that Vα ⊂ Uα for each α ∈ A (see [3, Theorem 1.4]). Thus the conclusion is obtained by repeating the proof of [11, Lemma 3.1 and Proposition 4.3]. 2 Applying [2, Theorem 3.1], we have the following. Lemma 2.2. Let X be a topological space and (Y, ρ) a metric space. Then every ρ-proximal continuous mapping Φ : X → 2Y satisfies the f.s.f.p. Proof. Let F be a closed subset of X and V a locally finite collection of open subsets of Y such that F ⊂ Φ−1 [ V ]. By virtue of [2, Theorem 3.1], there exists a σ-discrete (in X) collection W of cozero-sets of X such that W refines {Φ−1 [V ]: V ∈ V } and W = Φ−1 [ V ]. According to [9, Theorem 1.2] and [10, Theorem 1.2] (see also [3, Theorem 1.2]), the σ-discrete cozero-set cover {W ∩ F : W ∈ W } of F is refined by a locally finite cozero (in F ) cover U of F . Then U is the required cover. 2 The following lemma is easy to verify and we omit the proof. Lemma 2.3. Let X be a topological space, (Y, ρ) a metric space, Φ : X → F (Y ) a ρ-proximal continuous mapping and g : X → Y a continuous mapping. Then the mapping u : X → R defined by u(x) = ρ(g(x), Φ(x)) for x ∈ X is continuous. For Φ : X → 2Y , A ⊂ X and a selection h : A → Y for Φ|A , define Φh : X → 2Y by Φh (x) = {h(x)} if x ∈ A and Φh (x) = Φ(x) otherwise. The following technical theorem is a key to show Theorem 1.2. Theorem 2.4. Let X be a topological space, (Y, ρ) a metric space, Φ : X → 2Y a ρ-proximal continuous mapping, C a cozero-set of X, h : C → Y a continuous selection for Φ|C and Z a zero-set of X with Z ⊂ C. Also, let r > 0 and let g : X → Y be a continuous r-selection for Φh|Z . Then the mapping ϕ : X → 2Y defined by ϕ(x) = Φh|Z (x) ∩ B(g(x), r) for x ∈ X satisfies the f.s.f.p. Proof. Let F be a closed subset of X and V a locally finite open collection in Y such that F ⊂ −1 {ϕ [V ]: V ∈ V }. Since C is a cozero-set of X and ϕ(x) = {h(x)} for each x ∈ Z, h−1 ( V ) is a cozero set of X such that F ∩ Z ⊂ h−1 ( V ). Choose a cozero-set O of F such that F ∩ Z ⊂ O ⊂ O ⊂ h−1 ( V ) and put U0 = {h−1 (V ) ∩ O: V ∈ V }. Then U0 is a locally finite cozero-set collection of F such that U0
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refines {Φ−1 ∈ V } and F ∩ Z ⊂ U0 . Since F \ U0 is a zero-set of F and F \ Z is a cozero-set h|Z [V ]: V of F such that F \ U0 ⊂ F \ Z, there exist a cozero-set O of F and a closed subset E of F such that F \ U0 ⊂ O ⊂ E ⊂ F \ Z. We shall take a locally finite cozero-set (in E) cover G of E which refines {Φ−1 [V ]: V ∈ V }. Put U1 = {G ∩ O : G ∈ G }. Then U0 ∪ U1 is the required locally finite cozero-set (in F ) collection. Here, we show the existence of such a cover G of E. Note that Φh|Z (x) = Φ(x) for each x ∈ E since E ∩ Z = ∅. For each n ∈ N, put Un = x ∈ X: ρ g(x), Φ(x) < r − 1/n .
(2.1)
By Lemma 2.3, the collection {Un : n ∈ N} is a cozero-set cover of X. For each n ∈ N, take a locally finite open cover Wn of Y such that mesh Wn < 1/n. For each n ∈ N and W ∈ Wn , put n OW =
Φ−1 V ∩ B(W, r − 1/n) : V ∈ V
and
n GnW = Un ∩ g −1 (W ) ∩ OW .
(2.2) (2.3)
n n Since OW = Φ−1 [( V ) ∩ B(W, r − 1/n)], according to [2, Lemma 3.3], OW is a cozero-set of X. Thus, −1 n since {g (W ): W ∈ Wn } is a locally finite cozero-set cover of X, so is {GW : W ∈ Wn } for each n ∈ N. We claim that the collection {GnW : W ∈ Wn , n ∈ N} covers E. For, let x ∈ E. Then x ∈ ϕ−1 [V ] for some V ∈ V . Take y ∈ ϕ(x) ∩ V = Φ(x) ∩ B(g(x), r) ∩ V . Then ρ(g(x), y) < r. Choose n ∈ N satisfying ρ(g(x), y) < r − 1/n. Since Wn covers Y , we may take W ∈ Wn such that g(x) ∈ W . Then n . Since x ∈ g −1 (W ). Since y ∈ Φ(x) ∩ V ∩ B(W, r − 1/n), we have x ∈ Φ−1 [( V ) ∩ B(W, r − 1/n)] = OW n ρ(g(x), Φ(x)) ρ(g(x), y) < r − 1/n, we have x ∈ Un . Thus x ∈ GW . Fix n ∈ N and W ∈ Wn . We claim that there exists a σ-locally finite cozero-set (in E) collection UWn of E such that E ∩ GnW ⊂ UWn and UWn refines {Φ−1 [V ∩ B(W, r − 1/n)]: V ∈ V }. Indeed, since GnW is a cozero-set of X, we may take a countable collection {Ck : k ∈ N} of cozero-sets of E such that E ∩ GnW = k∈N Ck = k∈N Ck . For each k ∈ N, n E ∩ Ck ⊂ OW =
Φ−1 V ∩ B(W, r − 1/n) : V ∈ V
by (2.2) and (2.3). Since Φ satisfies the f.s.f.p. by Lemma 2.2, there exists a locally finite cozero-set (in E ∩ Ck ) cover Ck of E ∩ Ck such that Ck refines {Φ−1 [V ∩ B(W, r − 1/n)]: V ∈ V }. Then UWn = {C ∩ Ck : C ∈ Ck , k ∈ N} is the required σ-locally finite cozero-set (in E) collection. Put U = {U ∩ GnW : U ∈ UWn , W ∈ Wn , n ∈ N}. Then U is a σ-locally finite cozero-set (in E) cover of E. By virtue of [9, Theorem 1.2] and [10, Theorem 1.2] (see also [3, Theorem 1.2]), U is refined by a locally finite cozero-set (in E) cover G of E. To see that G is the required cover, it suffices to show that U refines {ϕ−1 [V ]: V ∈ V }. Let U ∈ U . Then U = U ∩ GnW for some U ∈ UWn , W ∈ Wn and n ∈ N. By the choice of UWn , we have U ⊂ Φ−1 [V ∩ B(W, r − 1/n)] for some V ∈ V . To see that U ⊂ ϕ−1 [V ], let x ∈ U = U ∩ GnW . By (2.1) and (2.3), we have ρ(g(x), Φ(x)) < r − 1/n and g(x) ∈ W . Also, we may take z ∈ Φ(x) ∩ V ∩ B(W, r − 1/n) because x ∈ U ⊂ Φ−1 [V ∩ B(W, r − 1/n)]. Choose y ∈ W such that ρ(y, z) < r − 1/n. Since mesh Wn < 1/n, we have ρ(g(x), y) < 1/n, and hence ρ(g(x), z) < r. Thus z ∈ Φ(x) ∩ B(g(x), r) ∩ V = ϕ(x) ∩ V , which implies x ∈ ϕ−1 [V ], and hence U ⊂ ϕ−1 [V ]. Therefore, ϕ satisfies the f.s.f.p. 2 Now, we prove Theorem 1.2. Proof of Theorem 1.2. By repeating the proof of (b) in [8, Theorem 2.1] with applying Lemma 2.1 and Theorem 2.4, we have the following:
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(∗) Let X be a topological space, Y a Banach space, α ∈ [0, 1), Φ : X → Fα (Y ) a ρ-proximal mapping, C a cozero-set of X, h : C → Y a continuous selection for Φ|C and Z a zero-set of X with Z ⊂ C. If there exists a continuous r-selection g : X → Y for Φh|Z for some r > 0, then there exists a continuous ∞ ˆ for all x ∈ X, where α ˆ = 1 + i=0 αi . selection f : X → Y for Φh|Z such that ρ(g(x), f (x)) < αr The statement (b) follows from (∗) with C = ∅. For (a), choose λ 2 such that ρ(0, Φ(x)) < λ for some x ∈ X, and let β = max{ˆ α, λ}. For each n ∈ N, put An = x ∈ X: ρ 0, Φ(x) β n−1 Bn = x ∈ X: ρ 0, Φ(x) < β n .
and
By Lemma 2.3, An is a zero-set and Bn is a cozero-set of X. Moreover, An ⊂ Bn ⊂ An+1 for n ∈ N and X = n∈N Bn . It suffices to construct continuous selections fn : Bn+1 → Y , n ∈ N, for Φ|Bn+1 such that fn+1 |Bn = fn |Bn for each n ∈ N. Then the mapping f : X → Y defined by f |Bn = fn |Bn for n ∈ N is a selection for Φ. Since {Bn : n ∈ N} is an open cover of X, f is continuous (see [6, Remark 2.4]). As in the proof of (a) in [8, Theorem 2.1], we construct the mappings fn , n ∈ N, by induction on n. To keep the induction going, we shall also require ρ(fn (x), 0) < β n+2 for every n ∈ N and x ∈ Bn+1 . The existence of f1 : B2 → Y follows from (b), with X replaced by B2 , r by β 2 , and with g(x) = 0 for every x ∈ A2 . Then ρ(f1 (x), 0) < α ˆ β 2 β 3 for every x ∈ B2 . Assume that fn : Bn+1 → Y has been constructed. n+2 for each x ∈ Bn+2 . By applying (∗) with replacing X by Bn+2 , C by Bn+1 , Then ρ(0, Φfn |An+1 (x)) < β n+2 h by fn , Z by An+1 , r by β , and with g(x) = 0 for every x ∈ Bn+2 , we have a continuous selection ˆ β n+2 β n+3 . Then fn+1 is also a selection fn+1 : Bn+2 → Y for Φfn |An+1 |Bn+2 such that ρ(fn+1 (x), 0) < α for Φ|Bn+2 . Since fn+1 |An+1 = fn |An+1 , we have fn+1 |Bn = fn |Bn . Thus fn+1 is the required mapping. 2 Remark 2.5. Semenov [16, Theorem 6] generalized Michael’s paraconvex-valued selection theorem (Theorem 1.1). Theorem 1.2 can be generalized similarly. For a non-empty closed subset P of a Banach space Y , r > 0 and y ∈ Y , we put δ P, B(y, r) = sup ρ(q, P )/r: q ∈ conv P ∩ B(y, r) , where we let δ(P, B(y, r)) = 0 if P ∩ B(y, r) = ∅. The function αP : (0, ∞) → [0, ∞) defined by αP (r) = sup δ P, B(y, r) : y ∈ Y is called the function of nonconvexity of P . Note that a closed subset P of Y is α-paraconvex if and only if αP (r) α for each r ∈ (0, ∞). A function β : (0, ∞) → [0, ∞) is said to be geometrically summable if for ∞ each t ∈ (0, ∞), the series n=0 qβn (t) converges, where the sequence {qβn (t)} is defined as follows: qβ0 (t) = t,
qβn (t) = β qβn−1 (t) · qβn−1 (t),
n ∈ N.
By an argument analogous to the proof of [16, Theorem 6] with applying Lemma 2.1 and Theorem 2.4, “Φ : X → Fα (Y )” in Theorem 1.2 can be relaxed to “Φ : X → F (Y ) with some geometrically summable function β : (0, ∞) → [0, ∞) such that αΦ(x) (t) < β(t) for each x ∈ X and t ∈ (0, ∞)”. 3. Other results on ρ-proximal continuous mappings The following corollary is immediate from Theorem 2.4.
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Corollary 3.1. Let X be a topological space, (Y, ρ) a metric space, Φ : X → 2Y a ρ-proximal continuous mapping, r > 0 and g : X → Y a continuous r-selection for Φ. Then the mapping ϕ : X → 2Y defined by ϕ(x) = Φ(x) ∩ B(g(x), r) for x ∈ X satisfies the f.s.f.p. Thus, by Lemma 2.1, we have the following. Corollary 3.2. Let X be a topological space, Y a Banach space and Φ : X → F0 (Y ) a ρ-proximal continuous mapping, r > 0 and g : X → Y a continuous r-selection for Φ. Then Φ admits a continuous selection f : X → Y such that ρ(g(x), f (x)) r for each x ∈ X. A mapping Ψ is said to be upper semicontinuous (u.s.c. for short) if for every closed subset F of Y , Φ−1 [F ] is closed in X. It is easy to see that every u.s.c. mapping is ρ-u.s.c. As in Lemma 2.1, by repeating the proof of [11, Lemma 3.1 and Proposition 4.1], we have the following lemma. Let C (Y ) denote the set of all non-empty compact subsets of Y . Lemma 3.3. Let X be a topological space, Y a completely metrizable space, Φ : X → F (Y ) a mapping with the f.s.f.p. Then there exist a u.s.c. mapping ψ : X → C (Y ) and an l.s.c. mapping ϕ : X → C (Y ) such that ϕ(x) ⊂ ψ(x) ⊂ Φ(x) for each x ∈ X. For a mapping Φ : X → 2Y , a mapping ϕ : X → 2Y is called a multi-selection for Φ if ϕ(x) ⊂ Φ(x) for each x ∈ X. By Corollary 3.1 and Lemma 3.3, we have the following corollary which answers a question asked by Gutev to the author. Corollary 3.4. Let X be a topological space, Y a completely metrizable space, Φ : X → F (Y ) a ρ-proximal continuous mapping, g : X → Y a continuous mapping and r > 0 such that ρ(g(x), Φ(x)) < r for all x ∈ X. Then there exist a u.s.c. multi-selection ψ : X → C (Y ) for Φ and an l.s.c. mapping ϕ : X → C (Y ) such that ϕ(x) ⊂ ψ(x) ⊂ B(g(x), r) for each x ∈ X. Acknowledgements The author would like to thank the referee for valuable comments and suggestions. References [1] R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. [2] V. Gutev, Weak factorizations of continuous set-valued mappings, Topol. Appl. 102 (2000) 33–51. [3] T. Hoshina, Extensions of mappings II, in: K. Morita, J. Nagata (Eds.), Topics in General Topology, North-Holland, Amsterdam, 1989, pp. 41–80. [4] G.E. Ivanov, Continuous selections of multifunctions with weakly convex values, Topol. Appl. 155 (2008) 851–857. [5] V. Klee, Circumspheres and inner products, Math. Scand. 8 (1960) 363–370. [6] N.R.L. Makala, Selections for paraconvex-valued mappings on non-paracompact domains, Topol. Appl. 159 (2012) 153–157. [7] E. Michael, Continuous selections I, Ann. Math. 63 (1956) 361–382. [8] E. Michael, Paraconvex sets, Math. Scand. 7 (1959) 372–376. [9] K. Morita, Paracompactness and product spaces, Fundam. Math. 50 (1962) 223–236. [10] K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964) 365–382. [11] S. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980) 291–317. [12] D. Repovš, P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht, 1998. [13] D. Repovš, P.V. Semenov, Continuous selections as uniform limits of δ-continuous ε-selections, Set-Valued Anal. 7 (1999) 239–254. [14] D. Repovš, P.V. Semenov, On continuous choice of retractions onto nonconvex subsets, Topol. Appl. 157 (2010) 1510–1517. [15] P.V. Semenov, Functionally paraconvex sets, Math. Notes 54 (1993) 1236–1240. [16] P.V. Semenov, Fixed-point theorems with controllable rejection of the convexity of the values of a multivalued mapping, Sb. Math. 189 (1998) 461–480. [17] P.V. Semenov, Nonconvexity in problems of multivalued calculus, J. Math. Sci. (N.Y.) 100 (2000) 2682–2699.