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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent:
(i) K is fragmented by $d_{D}$, where, for each S ⊂ D,
$d_{S}(x,y) = sup{ϱ(x(t),y(t)): t∈ S}$.
(ii) For each countable subset A of D, $(K,d_{A})$ is separable.i
(iii) The space (K,γ(D)) is Lindelöf, where γ(D) is the topology of uniform convergence on the family of countable subsets of D.
(iv) $(K,γ(D))^{{ℕ}}$ is Lindelöf.
The rest of the paper is devoted to applications of the basic theorem. Here are some of them. A compact Hausdorff space K is Radon-Nikodým compact if, and only if, there is a bounded subset D of C(K) separating the points of K such that (K,γ(D)) is Lindelöf. If X is a Banach space and H is a weak*-compact subset of the dual X* which is weakly Lindelöf, then $(H,{weak})^{ℕ}$ is Lindelöf. Furthermore, under the same condition $\overline{span(H)}^{|| ||}$ and $\overline{co(H)}^{w*}$ are weakly Lindelöf. The last conclusion answers a question by Talagrand. Finally we apply our basic theorem to certain classes of Banach spaces including weakly compactly generated ones and the duals of Asplund spaces.
(i) K is fragmented by $d_{D}$, where, for each S ⊂ D,
$d_{S}(x,y) = sup{ϱ(x(t),y(t)): t∈ S}$.
(ii) For each countable subset A of D, $(K,d_{A})$ is separable.i
(iii) The space (K,γ(D)) is Lindelöf, where γ(D) is the topology of uniform convergence on the family of countable subsets of D.
(iv) $(K,γ(D))^{{ℕ}}$ is Lindelöf.
The rest of the paper is devoted to applications of the basic theorem. Here are some of them. A compact Hausdorff space K is Radon-Nikodým compact if, and only if, there is a bounded subset D of C(K) separating the points of K such that (K,γ(D)) is Lindelöf. If X is a Banach space and H is a weak*-compact subset of the dual X* which is weakly Lindelöf, then $(H,{weak})^{ℕ}$ is Lindelöf. Furthermore, under the same condition $\overline{span(H)}^{|| ||}$ and $\overline{co(H)}^{w*}$ are weakly Lindelöf. The last conclusion answers a question by Talagrand. Finally we apply our basic theorem to certain classes of Banach spaces including weakly compactly generated ones and the duals of Asplund spaces.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
165-192
Opis fizyczny
Daty
wydano
2003
Twórcy
autor
- Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo Murcia, Spain
autor
- Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195-4350, U.S.A.
autor
- Departamento de Matemáticas, Universidad de Murcia, 30.100 Espinardo Murcia, Spain
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-2-4
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