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1
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The Lindelöf property and σ-fragmentability

100%
Cascales B., Namioka I.
Fundamenta Mathematicae
|
2003
|
tom 180
|
nr 2
161-183
EN
In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space $[-1,1]^{D}$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of $[-1,1]^{D}$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is the case then $(X,γ(D))^{ℕ}$ is also Lindelöf. We give several applications of this theorem in areas of topology and Banach spaces. We also show by examples that the main theorem cannot be extended to the cases where X is Čech-analytic and Lindelöf or countably K-determined.
2
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Fragmentability and σ-fragmentability

81%
Jayne J. E., Namioka I., Rogers C. A.
Fundamenta Mathematicae
|
1993
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tom 143
|
nr 3
207-220
EN
Recent work has studied the fragmentability and σ-fragmentability properties of Banach spaces. Here examples are given that justify the definitions that have been used. The fragmentability and σ-fragmentability properties of the spaces $ℓ^∞$ and $ℓ_c^∞({Γ})$, with Γ uncountable, are determined.
3
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σ-fragmented Banach spaces II

81%
E. Jayne J., Namioka I., Rogers C. A.
Studia Mathematica
|
1994
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tom 111
|
nr 1
69-80
EN
Recent papers have investigated the properties of σ-fragmented Banach spaces and have sought to find which Banach spaces are σ-fragmented and which are not. Banach spaces that have a norming M-basis are shown to be σ-fragmented using weakly closed sets. Zizler has shown that Banach spaces satisfying certain conditions have locally uniformly convex norms. Banach spaces that satisfy similar, but weaker conditions are shown to be σ-fragmented. An example, due to R. Pol, is given of a Banach space that is σ-fragmented using differences of weakly closed sets, but is not σ-fragmented using weakly closed sets.
4
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The Lindelöf property in Banach spaces

81%
Cascales B., Namioka I., Orihuela J.
Studia Mathematica
|
2003
|
tom 154
|
nr 2
165-192
EN
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.
5
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Complements of sets of unstable points

50%
Namioka I.
Fundamenta Mathematicae
|
1973
|
tom 80
|
nr 1
81-89
6
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On certain actions of semi-groups on L-spaces

40%
Namioka I.
Studia Mathematica
|
1966-1968
|
tom 29
|
nr 1
63-77
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