# Topological spaces compact with respect to a set of filters

Open Mathematics (2014)

- Volume: 12, Issue: 7, page 991-999
- ISSN: 2391-5455

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topPaolo Lipparini. "Topological spaces compact with respect to a set of filters." Open Mathematics 12.7 (2014): 991-999. <http://eudml.org/doc/269174>.

@article{PaoloLipparini2014,

abstract = {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.},

author = {Paolo Lipparini},

journal = {Open Mathematics},

keywords = {Filter; Ultrafilter convergence; Sequencewise -compactness; Preservation under products; Comfort pre-order; Sequential compactness; ultrafilter convergence; sequencewise -compactness; sequential compactness; comfort pre-order},

language = {eng},

number = {7},

pages = {991-999},

title = {Topological spaces compact with respect to a set of filters},

url = {http://eudml.org/doc/269174},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Paolo Lipparini

TI - Topological spaces compact with respect to a set of filters

JO - Open Mathematics

PY - 2014

VL - 12

IS - 7

SP - 991

EP - 999

AB - 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.

LA - eng

KW - Filter; Ultrafilter convergence; Sequencewise -compactness; Preservation under products; Comfort pre-order; Sequential compactness; ultrafilter convergence; sequencewise -compactness; sequential compactness; comfort pre-order

UR - http://eudml.org/doc/269174

ER -

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