THE SPACES OF CLOSED CONVEX SETS IN EUCLIDEAN SPACES WITH THE FELL TOPOLOGY KATSURO SAKAI AND ZHONGQIANG YANG Abstract. Let ConvF (Rn ) be the space of all non-empty closed convex sets in Euclidean space Rn endowed with the Fell topology. In this paper, we prove that ConvF (Rn ) ≈ Rn × Q for every n > 1 whereas ConvF (R) ≈ R × I.
Let Conv(X) be the set of all non-empty closed convex sets in a normed linear space X = (X, k·k). We can consider various topologies on Conv(X). In the paper [6], the AR-property of the spaces Conv(X) with the Hausdorff metric topology, the Attouch-Wets topology, and the Wijsman topology has been studied. In this paper, we shall consider the Fell topology on Conv(X), which is generated by the sets of the form U − = {A ∈ Conv(X) | A ∩ U 6= ∅} and (X \ K)+ = {A ∈ Conv(X) | A ⊂ X \ K}, where U is open and K is compact in X. This topology is also defined on the set Conv∗ (X) = Conv(X) ∪ {∅}. By Conv∗F (X) and ConvF (X), we denote the spaces Conv∗ (X) and Conv(X) admitting the Fell topology. In case X is finite-dimensional (equivalently locally compact), ConvF (X) is a locally compact metrizable space and Conv∗F (X) is its Alexandorff onepoint compactification. It is easy to see that ConvF ((0, 1)) is homeomorphic to (≈) the triangle with two vertices removed, ∆ \ {(0, 0), (1, 1)}, where ∆ = {(x, y) ∈ I2 | x 6 y} ⊂ I2 . Since ConvF (R) ≈ ConvF ((0, 1)), we have ConvF (R) ≈ ∆ \ {(0, 0), (1, 1)} ≈ R × I, hence it follows that Conv∗F (R) ≈ ∆/{(0, 0), (1, 1)} ≈ (S1 × I)/({pt} × I), 1991 Mathematics Subject Classification. 54B20, 54D05, 54E45, 57N20. Key words and phrases. The hyperspace, closed convex sets, Euclidean space, the Fell topology, the Hilbert cube. The first author was supported for this work by Grant-in-Aid for Scientific Reserch (No. 17540061), Japan Society for the Promotion of Science. The second author was supported for this work by National Natural Science Foundation of China (No. 10471084) and by Guangdong Provincial Natural Science Foundation (No. 04010985). 1
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where S1 is the unit circle. For n > 1, the space ConvF (Rn ) is infinitedimensional. Let Q = [−1, 1]N be the Hilbert cube. We prove the following result: Main Theorem. For each n > 1, ConvF (Rn ) ≈ Rn × Q and Conv∗F (Rn ) ≈ (Sn × Q)/({pt} × Q) ≈ (Bn × Q)/(Sn−1 × Q), where Bn and Sn−1 are the closed unit ball and the unit sphere in Rn . Remark 1. As studied in [6], Conv(X) has other metrizable topologies called the Attouch-Wets topology and the Wijsman topology. However, in case X is finite-dimensional, these are equal to the Fell topology. For such topologies, refer to the book [1]. Remark 2. The space ConvH (X) with the Hausdorff metric topology is rather complicated. Concerning the subspace CCH (X) ⊂ ConvH (X) consisting of non-empty compact convex sets, it is shown in [4] in case n > 1 that CCH (Rn ) ≈ Q\{0}. It should be remarked that CCF (Rn ) = CCH (Rn ), which can be obtained by [8, Theorem 3]. As is observed in [6, §2], CCH (Rn ) is a component of ConvH (Rn ).1 However, as will be seen in Proposition 3, CCF (Rn ) is homotopy dense in ConvH (Rn ). The open ball and the closed ball in Rn centered at the point x ∈ Rn with radius r > 0 are respectively denoted as follows: B(x, r) = int(x + rBn ) and B(x, r) = x + rBn . Proposition 1. For every n ∈ N, Conv∗F (Rn ) is compact, hence it is the Alexandorff one-point compactification of ConvF (Rn ). Proof. Since the hyperspace Cld∗F (Rn ) of all closed sets in Rn with the Fell topology is compact [1, Theorem 5.1.3], it suffices to show that Conv∗F (Rn ) is closed in Cld∗F (Rn ). For A ∈ Cld∗ (Rn ) \ Conv∗ (Rn ), we have a, b ∈ A and c ∈ ha, bi \ A, where ha, bi is the convex hull of {a, b}. Choose ε > 0 and δ > 0 such that B(c, ε) ∩ A = ∅ and hx, yi ∩ B(c, ε) 6= ∅ if kx − ak < δ and ky − bk < δ. Then, (Rn \ B(c, ε))+ ∩ B(a, δ)− ∩ B(b, δ)− is a neighborhood of A which misses Conv∗ (Rn ).
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Every locally compact Hausdorff space X has the Alexandorff one-point compactification, which is denoted by αX = X ∪ {∞}. Let f : X → Y be a map between locally compact Hausdorff spaces. If f is proper, that is, f −1 (C) is compact for each compact set C ⊂ Y , then f extends to the map f˜ : αX → αY such that f˜(∞) = ∞. By identifying X with the 1The
n n subspace ConvB H (R ) ⊂ ConvH (R ) consisting of all bounded closed convex sets is coincides with CCH (Rn ).
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subset of CldF (X) consisting of singletons and ∞ with ∅, we can regard αX ⊂ Cld∗F (X). For A ∈ Conv(Rn ), let p(A) be the nearest point of A from the origin 0 ∈ Rn with respect to the Euclidean metric (cf. the proof of [5, Lemma 1.6]). Lemma 2. The function p : ConvF (Rn ) → Rn is continuous and proper, hence it extends to the map p∗ : Conv∗F (Rn ) → αRn by p∗ (∅) = ∞. Proof. For each ε > 0, A ∈ Conv(Rn ) has the following neighborhood: U = B(p(A), ε)− ∩ (Rn \ (kp(A)k − ε)Bn )+ ∩ Conv(Rn ), where (kp(A)k − ε)Bn = ∅ if kp(A)k − ε < 0. Then, for every B ∈ U, kp(A)k − ε < kp(B)k < kp(A)k + ε, which implies kp(A) − p(B)k < ε. Hence, p is continuous at A. For each r > 0, p−1 (rBn ) is a closed subset of ConvF (Rn ) \ (Rn \ rBn )+ = Conv∗F (Rn ) \ (Rn \ rBn )+ , which is compact by Proposition 1. Then, p−1 (rBn ) is also compact. It follows that p is proper. ¤ Proposition 3. There is a homotopy h : Conv∗F (Rn ) × I → Conv∗F (Rn ) such that h0 = id, h1 = p∗ , ht |αRn = id and p∗ ht = p∗ for every t ∈ I, h({∅} × I) = {∅} and h(Conv(Rn ) × (0, 1]) ⊂ CC(Rn ). Thus, αRn (resp. Rn ) is a strong deformation retract of Conv∗F (Rn ) (resp. ConvF (Rn )), CC∗ (Rn ) (resp. CC(Rn )) is homotopy dense in Conv∗F (Rn ) (resp. ConvF (Rn )) and each fiber of p∗ is contractible (hence p∗ is a CEmap). Proof. The desired homotopy h is defined as follows: h0 = id, h({∅} × I) = {∅} and ¶ 1−t n ht (A) = A ∩ p(A) + B for A ∈ Conv(Rn ) and t > 0. t µ
Obviously, h satisfies the conditions. It remains to verify the continuity of h. Since p(ht (A)) = p(A) for all A ∈ Conv(Rn ) and t ∈ I, h−1 ((Rn \ rBn )+ ) = (Rn \ rBn )+ × I for r > 0, hence h is continuous at (∅, t). Let A ∈ Conv(Rn ) and t ∈ I. Assume that K ⊂ Rn is compact and ht (A) ∩ K = ∅. When t = 0, V = (Rn \ K)+ ∩ Conv(Rn ) is a neighborhood
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of A in ConvF (Rn ) and hs (B) ∩ K = ∅ for all B ∈ V and s ∈ I. In case t > 0, choose 0 < ² < t/2 so that ¶ µ 1 − t + 2² n B = ∅. K ∩ A ∩ p(A) + t − 2² Since p is continuous, A has a neighborhood U in Conv(Rn ) such that B ∈ U implies 1 − t + 2² 1 − t + ² kp(A) − p(B)k < − , t − 2² t−² and then for s > t − ², 1−s n 1−t+² n 1 − t + 2² n p(B) + B ⊂ p(B) + B ⊂ p(A) + B . s t−² t − 2² Thus, A has the following neighborhood in ConvF (Rn ): µ µ µ ¶¶¶+ 1 − t + 2² n−1 n V = U ∩ R \ K ∩ p(A) + B . t − 2² Then, hs (B) ∩ K = ∅ for every B ∈ V and s > t − ². Next, assume that U ⊂ Rn is open and ht (A) ∩ U 6= ∅. When t = 1, p(A) ∈ U . By continuity of p, V = p−1 (U ) is a neighborhood of A in ConvF (Rn ), and p(B) ∈ hs (B) ∩ U for all B ∈ V. In case t < 1, choose 0 < ² < (1 − t)/2 so that µ ¶ 1 − t − 2² n U ∩ A ∩ p(A) + B 6= ∅. t + 2² We have a neighborhood U of A in ConvF (Rn ) such that B ∈ U implies 1 − t − ² 1 − t − 2² kp(A) − p(B)k < − , t+² t + 2² and then for s < t + ², 1 − t − 2² n 1−t−² n 1−s n p(A) + B ⊂ p(B) + B ⊂ p(B) + B . t + 2² t+² s Then, V = U ∩ U − is a neighborhood of A in ConvF (Rn ) and hs (B) ∩ U 6= ∅ for every B ∈ V and s < t + ². ¤ A separable metrizable space M is called a Hilbert cube manifold or a Q-manifold if each point of M has an open neighborhood which is homeomorphic to an open set in Q. Corollary 4. For every n > 1, ConvF (Rn ) is a Q-manifold. Proof. As observed in Remark 2, CCF (Rn ) = CCV (Rn ) ≈ Q \ {0} for every n > 1. Since CCF (Rn ) is homotopy dense in ConvF (Rn ) by Proposition 3, we can apply the Toru´ nczyk Characterization of Q-manifolds [9] to show that ConvF (Rn ) is a Q-manifold. ¤
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Now, we prove the Main Theorem. Proof of Main Theorem. First, note that Rn × Q is a Q-manifold. Since p is a CE-map by Proposition 3, p × id : ConvF (Rn ) × Q → Rn × Q is a near homeomorphism by the CE Approximation Theorem [2, 43.1]. By the Stability Theorem [2, 15.1], ConvF (Rn ) × Q ≈ ConvF (Rn ).2 Then, it follows that ConvF (Rn ) ≈ Rn × Q. Moreover, by Proposition 1, we have Conv∗F (Rn ) ≈ α(Rn × Q) ≈ (Sn × Q)/({pt} × Q). The proof is completed.
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The following is the direct consequence of the above proof: Corollary 5. For each n ∈ N, Conv∗F (Rn ) has the unique singular point ∅ and Conv∗F (Rn ) has the homotopy type of Sn . If m 6= n then neither Conv∗F (Rn ) ≈ Conv∗F (Rm ) nor ConvF (Rn ) ≈ ConvF (Rm ). ¤ References [1] G. Beer, Topologies on Closed and Closed Convex Sets, MAIA 268, Kluwer Acad. Publ., Dordrecht, 1993. [2] T.A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, Amer. Marh. Soc., Providence, RI, 1976. [3] J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, NorthHolland Math. Library 43, Elsevier Sci. Publ. B.V., Amsterdam, 1989. [4] S.B. Nadler, Jr., J. Quinn and N.M. Stavrakas, Hyperspaces of compact convex sets, Pacific J. Math. 83 (1979), 441–462. [5] Nguyen To Nhu, K. Sakai and R.Y. Wong, Spaces of retractions which are homeomorphic to Hilbert space, Fund. Math. 136 (1990), 45–52. [6] K. Sakai and M. Yaguchi, The AR-property of the spaces of closed convex sets, Colloq. Math. 106 (2006), 15–24. [7] K. Sakai and Z. Yang, Hyperspaces of non-compact metrizable space which are homeomorphic to the Hilbert cube, Topolog Appl. 127 (2002), 331–342. [8] Z. Yang and K. Sakai, The space of limits of continua in the Fell topology, Houston J. Math. 29 (2003), 325–335. [9] H. Toru´ nczyk, On CE-images of the Hilber cube and characterizations of Qmanifolds, Fund. Math. 106 (1980), 31–40. (Katsuro Sakai) Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan E-mail address:
[email protected] (Zhongqiang Yang) Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China P.R. E-mail address:
[email protected]
2For non-compact Q-manifolds, the book [3] is not sufficient one should refer to Chapman’s Lecture Notes [2].
