H-closed extensions with countable remainder
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 123-137
Voir la notice de l'article provenant de la source Czech Digital Mathematics Library
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.
@article{CMUC_2012__53_1_a8,
author = {McNeill, Daniel K.},
title = {H-closed extensions with countable remainder},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {123--137},
publisher = {mathdoc},
volume = {53},
number = {1},
year = {2012},
mrnumber = {2880915},
zbl = {1249.54047},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a8/}
}
McNeill, Daniel K. H-closed extensions with countable remainder. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 123-137. http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a8/