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Set Convergence and the Class of Compact Subsets

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Set Convergence and the Class of Compact Subsets ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ ”AL.I.CUZA” IAS¸I Tomul XLVII, s.I a, Matematic˘a,2001, f.2. SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS BY G. APREUTESEI 1. Introduction. If (X, d) is a metric space and Cl(X) is the class of closed subsets of X, one can defined on Cl(X) some topologies or conver- gences, called hypertopologies or hyperconvergences. In recent years, many efforts have been done in order to find some conditions concerning the space (X, d) which assure the coincidence of some known hypertopologies or hyperconvergences on the fixed class Cl(X) ([7], [6], [11] etc.); if now the space (X, d) is fixed, an other problem, initiated in [1], is to specify some subclasses (even ”the best” subclass) A of Cl(X) on which two precised hypetopologies (or hyperconvergences) coincide. In [1], the author establish that on the class K(X) of the compact sets of X, many hypertopologies coincide, more precisely K(X) P ≡ V ≡ `f ≡ H ⇓ bP ≡ AW where the topologies V, lf, H, P, AW, bP will be described in the Section 2. In this paper, we offer new results concerning the class K(X) with re- spect to Pompeiu-Hausdorff and Wijsman convergences for monotone se- quences (Section 3, Theorem 3.1), as well as Mosco and Pompeiu-Hausdorff convergences of nets (Section 4, Theorem 4.1, which is formulated in terms of necessary and sufficient condition). 264 G. APREUTESEI 2 2. Preliminaries and notations. Let (X, d) be a metric space. A distance ρ is called equivqlently to d if they induce the same topology on X. We denote by D the family of all metrics that are equivalent to d. Also we give the notations: B(X) = {A ∈ Cl(X); A is d-bounded }; S(a, ε) = {x ∈ X; d(a, x) < ε} with a ∈ X, ε > 0 –the ball with center a and radius ε; B(a, ε) = {x ∈ X; d(a, x) ≤ ε} with a ∈ X, ε > 0 –the closed ball with center a and radius ε; [ Sε(A) = S(a, ε), ε–enlargement of the set A; a∈A cl A = the d-closure of the set A. In order to define the desired convergences, we introduce: i) the Hausdorff excess of A with respect to B e(A, B) = sup{d(a, B); a ∈ A}; ii) the r-excess of A with respect to B er(A, B) = e(A ∩ B(x0, r),B), where x0 is fixed in X and r is a positive real number. iii) for a net (Ai)i∈I of closed subsets of X (where I is a nonvoid directed index set), the lower limit , respectively the upper limit of (Ai) are defined as follows lim inf(Ai) = {a ∈ X; ∀i ∈ I ∃ ai ∈ Ai such that a = lim ai} = {a ∈ X; lim d(a, Ai) = 0}, respectively lim sup(Ai) = {a ∈ X; ∃ J ⊂ I cofinally in I such that ∀j ∈ J ∃ aj ∈ Aj with a = lim aj} \ [ = cl Ai . i∈I j≥i If X is a linear topological space, we denote the topological dual of X by X∗ and the weak topology on X by w. 3 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 265 Also, C(X) = {A ∈ Cl(X); A is convex} and Ac the intersection of C(X) with A. The support function of a closed convex set A is s(x∗,A) = sup{x∗(a); a∈A}, where x∗ ∈X∗. In many cases, it will be convenient to consider ([17]) a hyperconver- gence c as the conjunction of a ”lower convergence” (c−) and an ”upper convergence” (c+). We consider X a metric space. The Pompeiu-Hausdorff convergence is H, the conjunction of H− and H+, where (H−) lim e(A, Ai) = 0, (H+) lim e(Ai,A) = 0. If denoted by H(A, B) = max{e(A, B), e(B, A)}, then (Ai) is H - con- vergent to A iff lim H(Ai,A) = 0. The Attouch-Wets convergence is AW , the conjunction of AW− and AW+, where (AW−) lim er(A, Ai) = 0, (AW+) lim er(Ai,A) = 0, for every positive real r and arbitrary x0. The Wijsman convergence is W , the conjunction of W− and W+, where (W−) ∀x∈X, d(x, A) ≥ lim sup d(x, Ai), (W+) ∀x∈X, d(x, A) ≤ lim inf d(x, Ai). The Kuratowski convergence is K, the conjunction of K− and K+, where (K−) A ⊂ lim inf(Ai), (K+) lim sup(Ai) ⊂ A. 266 G. APREUTESEI 4 We notice that K− is equivalently to W−. The proximal convergence is P , the conjunction of H+ and K−. The b-proximal convergence is bP , the conjunction of AW+ and K−. The locally finite convergence is lf, the supremum of all conver- gences H(d), when d ∈ D ([7]). The Vietoris convergence is V , the supremum of all convergences P (d), when d ∈ D ([7]). The Mosco convergence is M,the conjunction of K− and M+ (see [12], [2]), where (M+) w − lim sup(Ai) ⊂ A, and the linear convergence is L, the conjunction of K− and S+, where ∗ ∗ ∗ ∗ (S+) ∀x ∈X , lim sup s(x ,Ai) ≤ s(x ,A). We note that all this convergence, except the Kuratowski and Mosco convergences, are topological. The relations between they are given on K(X) by the table of first page. On Cl(X) or C(X) we have also the impli- cations ([16] and [17]): P =⇒ L =⇒ M =⇒ K; AW =⇒ W =⇒ K. Finally, we recall that a family A of closed sets of a metric space (X, d) is stable with respect to closed subsets if for any set A ∈ A and B ⊂ A with B ∈ Cl(X) it follows that B ∈ A. The Wijsmann convergence for the monotone sequence. In the following, we intent to observe if the convergences (or the topologies) Wijsman and Hausdorff agree with on K(X). This is not true, as we observe from the followind example: Example 3.1. Lets consider `1 with the usual norm. ∗ We take An = {x ∈ `1; x = ten, t ∈ [0, 1]}, ∀n ∈ IN , where by en we denote the unit vector of the canonical base in `1, and A = {0}. Obviously, ∗ the sets A and An, n ∈ IN are compact. We observe that A = τW −lim An. Let be now x = (ξ1, ξ2, ..., ξn, ...) ∈ `1 arbitrarily fixed; we evaluate the distance between x and An: 5 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 267 Let be n ∈ IN sufficiently large such that |ξn| ≤ 1. Then d(x, An) = inf{kx − tenk; t ∈ [0, 1]} = ∞ X = inf |ξk| + |ξn − t|; t ∈ [0, 1] = k=1 k6=n ( ∞ ) X = inf |ξk| − |ξn| + |ξn − t|; t ∈ [0, 1] = k=1 = kxk − |ξn| + inf{|ξn − t|; t ∈ [0, 1]} = = kxk − |ξn| + max{−ξn, 0}. So, it follows that d(x, An) → kxk = d(x, A). Now, if we consider sup{d(x, An) − d(x, A); x ∈ B(0, 1)} ≥ |d(en,An) − d(en,A)| = 1, so A 6= AW − lim An. Hence, A 6= τH − lim An. Still the problem has a positive answer for the monotone sequences from K(X). Theorem 3.1. Let (X, d) be a metric space. (i) If (An)n∈IN∗ ⊂ Cl(X) with A1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · · and A ∈ K(X) such that A = τW − lim An, then An ⊂ A ( so An are compact) ! [ A = cl An and A = τH − lim An. n∈IN (ii) If (An)n∈IN∗ ⊂ K(X) with A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · and A ∈ Cl(X) such that A = τW − lim An, then A ⊂ An, A are compact, \ A = An and A = τH − lim An. n∈IN ∗ Proof. i) Let be k ∈ IN arbitrarily fixed and u ∈ Ak. Then u ∈ An+k, ∗ ∀n ∈ IN, so d(u, An+k) = 0, ∀n ∈ IN . As A = τW − lim An it follows that d(u, A) = lim d(u, An+k), so d(u, A) = 0. From the closure property of A , n→∞ ∗ we find that u ∈ A, i.e. Ak ⊂ A, ∀k ∈ IN , and Ak are compact. Now: e(An,A) = sup{d(x, A); x ∈ An} = 0. 268 G. APREUTESEI 6 From An ⊂ An+1 we have that d(a, An) ≥ d(a, An+1), ∀a ∈ A and then ∗ e(A, An) ≥ e(A, An+1). We denote by αn = e(A, An), n ∈ IN and by α ≥ 0 the limit of the decreasing sequence pozitive numbers (αn)n∈IN∗ . ∗ The function d(·,An) is continuous, ∀n ∈ IN and A is compact. It follows that ∃ an ∈ An such that αn = sup{d(a, An); a ∈ A} = d(an,An). ∗ ∗ From the compactness of A we find a subsequence (ank )k∈IN ⊂ (an)n∈IN which is convergent to an element a0 ∈ A. We have: 0 ≤ d(ank ,Ank ) ≤ d(ank , a0) + d(a0,Ank ). But d(ank , a0)−→0, d(a0,Ank )−→d(a0,A) from the τW –convergence of the ∗ sequence (An)n∈IN la A, and so αnk = d(ank ,Ank )−→0, that is α = 0. As e(An,A) = 0, we obtain that A = τH − lim An. ii) If u ∈ X, from An ⊃ An+1 we find that d(u, An) ≤ d(u, An+1). Partic- ularly, the sequence (d(a, An))n∈IN∗ of positive numbers is increasing ∀a ∈ A. From the τW –convergence of An to A it follows that d(a, An)−→d(a, A) = ∗ 0, and then d(a, An) = 0, i.e. a ∈ An, ∀n ∈ IN , though A ⊂ An (and A is compact). e(A, An) = sup{d(a, An); a ∈ A} = 0.
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