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By Harold R. (-txt) Byerly and Lutzer David (-cwm


The theory of metric spaces is closely related to the study of conditions that guarantee a topological space to be metrizable. In this direction the notions of c-semi-stratifiable (CSS) space, Gδ subset and Gδ-diagonal play a central role. In this paper the authors go more deeply into the relationship between the mentioned notions. In particular they give sufficient conditions for the metrizability of countable compact subsets of a topological space at the same time that they provide sufficient conditions for a compact subset of a topological space to be a Gδ-set. Hence, the authors prove that if the topological space is Hausdorff with a δθ-base (either with a point-countable T1-pointseparating open cover or with a quasi-Gδ-diagonal) then each countable compact subset is a compact metrizable Gδ-subset. In addition, conditions for a local CSS space to be a CSS space are given by using a novel characterization of CSS spaces in terms of a special convergent sequence, which is related, in one sense, to the notion of α-functions [R. E. Hodel, Pacific J. Math. 38 (1971), 641–652; MR0307169 (46 #6290)] and Θ-functions [P. Fletcher and W. F. Lindgren, Pacific J. Math. 71 (1977), no. 2, 419–428; MR0493968 (58 #12919)]. Furthermore, the mentioned characterization improves in a sense the one obtained by Fletcher and Lindgren for Hausdorff spaces in terms of Θ-functions [op. cit.]. However, the special relevance of this wor

Year: 2013
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