H-closed extensions with countable remainder

Daniel K. McNeill

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 1, page 123-137
  • ISSN: 0010-2628

Abstract

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This paper investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case is more difficult, however, we arrive at a necessary condition — a generalization of Čech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Császár, we show that a space X is a Čech g -space if and only if X is G δ in σ X or equivalently if E X is Čech complete. An example of a space which is a Čech f -space but not a Čech g -space is given answering a couple of questions of Császár. We show that if X is a Čech g -space and R ( E X ) , the residue of E X , is Lindelöf, then X has an H-closed extension with countable remainder. Finally, we investigate some natural generalizations of the residue to the class of all Hausdorff spaces.

How to cite

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McNeill, Daniel K.. "H-closed extensions with countable remainder." Commentationes Mathematicae Universitatis Carolinae 53.1 (2012): 123-137. <http://eudml.org/doc/246731>.

@article{McNeill2012,
abstract = {This paper investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case is more difficult, however, we arrive at a necessary condition — a generalization of Čech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Császár, we show that a space $X$ is a Čech $g$-space if and only if $X$ is $G_\delta $ in $\sigma X$ or equivalently if $EX$ is Čech complete. An example of a space which is a Čech $f$-space but not a Čech $g$-space is given answering a couple of questions of Császár. We show that if $X$ is a Čech $g$-space and $R(EX)$, the residue of $EX$, is Lindelöf, then $X$ has an H-closed extension with countable remainder. Finally, we investigate some natural generalizations of the residue to the class of all Hausdorff spaces.},
author = {McNeill, Daniel K.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Čech complete; H-closed; extension; -closed space; remainder; countable space; Čech -space; Katetov space},
language = {eng},
number = {1},
pages = {123-137},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {H-closed extensions with countable remainder},
url = {http://eudml.org/doc/246731},
volume = {53},
year = {2012},
}

TY - JOUR
AU - McNeill, Daniel K.
TI - H-closed extensions with countable remainder
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 1
SP - 123
EP - 137
AB - This paper investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case is more difficult, however, we arrive at a necessary condition — a generalization of Čech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Császár, we show that a space $X$ is a Čech $g$-space if and only if $X$ is $G_\delta $ in $\sigma X$ or equivalently if $EX$ is Čech complete. An example of a space which is a Čech $f$-space but not a Čech $g$-space is given answering a couple of questions of Császár. We show that if $X$ is a Čech $g$-space and $R(EX)$, the residue of $EX$, is Lindelöf, then $X$ has an H-closed extension with countable remainder. Finally, we investigate some natural generalizations of the residue to the class of all Hausdorff spaces.
LA - eng
KW - Čech complete; H-closed; extension; -closed space; remainder; countable space; Čech -space; Katetov space
UR - http://eudml.org/doc/246731
ER -

References

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