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compact space; countably compact space; Lindelöf space; $\Cal K$-starcompact space; $\Cal C$-starcompact space; $\Cal L$-starcompact space
A space $X$ is {\it $\Cal C$-starcompact} if for every open cover $\Cal U$ of $X,$ there exists a countably compact subset $C$ of $X$ such that $\mathop{\rm St}(C,{\Cal U})=X.$ In this paper we investigate the relations between $\Cal C$-starcompact spaces and other related spaces, and also study topological properties of $\Cal C$-starcompact spaces.
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