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Keywords:
Čech complete; H-closed; extension
Čech complete; H-closed; extension
Summary:
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.
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