Compact coverings for Baire locally convex spaces |
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Authors: | J Ka?kol |
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Institution: | a Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland b Departamento de Matemática Aplicada and Impa, Universidad Politécnica, E-46022 Valencia, Spain |
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Abstract: | Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space Cp(X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff Cp(X) is Baire (i.e. of the second Baire category) and is covered by a family of compact sets such that Kα⊂Kβ if α?β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete. |
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Keywords: | Baire spaces Compact coverings Fré chet spaces Locally convex spaces and metrizability |
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