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2014 | 12 | 7 | 991-999
Tytuł artykułu

Topological spaces compact with respect to a set of filters

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If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T 1 topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.
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Bibliografia
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  • [17] Lipparini P., A characterization of the Menger property by means of ultrafilter convergence, Topology Appl., 2013, 160(18), 2505–2513 http://dx.doi.org/10.1016/j.topol.2013.07.044
  • [18] Lipparini P., Productivity of [µ; λ]-compactness, preprint available at http://arxiv.org/abs/1210.2121
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bwmeta1.element.doi-10_2478_s11533-013-0398-2
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