Topological spaces compact with respect to a set of filters

Paolo Lipparini

Open Mathematics, Tome 12 (2014), p. 991-999 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2014-01-01

Zbl 1297.54011

EUDML-ID : urn:eudml:doc:269174

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