A very general covering property

Paolo Lipparini

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 2, page 281-306
  • ISSN: 0010-2628

Abstract

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We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are shown to be equivalent to a covering property in the sense considered here (Corollary 3.10). Conversely, every covering property is equivalent to the existence of appropriate kinds of accumulation points for arbitrary sequences on some fixed index set (Corollary 3.5). We discuss corresponding notions related to sequential compactness, and to pseudocompactness, or, more generally, properties connected with the existence of limit points of sequences of subsets. In spite of the great generality of our treatment, many results here appear to be new even in very special cases, such as D -compactness and D -pseudocompactness, for D an ultrafilter, and weak (quasi) M -(pseudo)-compactness, for M a set of ultrafilters, as well as for [ β , α ] -compactness, with β and α ordinals.

How to cite

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Lipparini, Paolo. "A very general covering property." Commentationes Mathematicae Universitatis Carolinae 53.2 (2012): 281-306. <http://eudml.org/doc/246390>.

@article{Lipparini2012,
abstract = {We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are shown to be equivalent to a covering property in the sense considered here (Corollary 3.10). Conversely, every covering property is equivalent to the existence of appropriate kinds of accumulation points for arbitrary sequences on some fixed index set (Corollary 3.5). We discuss corresponding notions related to sequential compactness, and to pseudocompactness, or, more generally, properties connected with the existence of limit points of sequences of subsets. In spite of the great generality of our treatment, many results here appear to be new even in very special cases, such as $D$-compactness and $D$-pseudocompactness, for $D$ an ultrafilter, and weak (quasi) $M$-(pseudo)-compactness, for $M$ a set of ultrafilters, as well as for $[\beta ,\alpha ]$-compactness, with $\beta $ and $\alpha $ ordinals.},
author = {Lipparini, Paolo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {covering property; subcover; compactness; accumulation point; convergence; pseudocompactness; limit point; covering property; subcover; compactness; accumulation point; pseudocompactness; limit point},
language = {eng},
number = {2},
pages = {281-306},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A very general covering property},
url = {http://eudml.org/doc/246390},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Lipparini, Paolo
TI - A very general covering property
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 2
SP - 281
EP - 306
AB - We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are shown to be equivalent to a covering property in the sense considered here (Corollary 3.10). Conversely, every covering property is equivalent to the existence of appropriate kinds of accumulation points for arbitrary sequences on some fixed index set (Corollary 3.5). We discuss corresponding notions related to sequential compactness, and to pseudocompactness, or, more generally, properties connected with the existence of limit points of sequences of subsets. In spite of the great generality of our treatment, many results here appear to be new even in very special cases, such as $D$-compactness and $D$-pseudocompactness, for $D$ an ultrafilter, and weak (quasi) $M$-(pseudo)-compactness, for $M$ a set of ultrafilters, as well as for $[\beta ,\alpha ]$-compactness, with $\beta $ and $\alpha $ ordinals.
LA - eng
KW - covering property; subcover; compactness; accumulation point; convergence; pseudocompactness; limit point; covering property; subcover; compactness; accumulation point; pseudocompactness; limit point
UR - http://eudml.org/doc/246390
ER -

References

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