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 \v Cech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Cs\'asz\'ar, we show that a space $X$ is a \v Cech $g$-space if and only if $X$ is $G_\delta$ in $\sigma X$ or equivalently if $EX$ is \v Cech complete. An example of a space which is a \v Cech $f$-space but not a \v Cech $g$-space is given answering a couple of questions of Cs\'asz\'ar. We show that if $X$ is a \v Cech $g$-space and $R(EX)$, the residue of $EX$, is Lindel\" of, 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.
Keywords: \v Cech complete, H-closed, extension
AMS Subject Classification: 54A25 54D35 54D40