Topological spaces compact with respect to a set of filters

Paolo Lipparini

Open Mathematics (2014)

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

Abstract

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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.

How to cite

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Paolo 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 -

References

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