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countably compact space; almost countably compact space; relatively almost countably compact subset
A subset $Y$ of a space $X$ is almost countably compact in $X$ if for every countable cover $\Cal U$ of $Y$ by open subsets of $X$, there exists a finite subfamily $\Cal V$ of $\Cal U$ such that $Y\subseteq \overline {\bigcup \Cal V}$. In this paper we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and also study various properties of relatively almost countably compact subsets.
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