A space X is C-starcompact if for every open cover U of X, there exists a countably compact subset C of X such that St(C,U) = X. In this paper we investigate the relations between C-starcompact spaces and other related spaces, and also study topological properties of C-starcompact spaces.
A subset Y of a space X is almost countably compact in X if for every countable cover U of Y by open subsets of X, there exists a finite subfamily V of U such that Y ⊆ U 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.
In this paper, we prove the following statements: \item {(1)} There exists a Hausdorff Lindelöf space $X$ such that the Alexandroff duplicate $A(X)$ of $X$ is not discretely absolutely star-Lindelöf. \item {(2)} If $X$ is a regular Lindelöf space, then $A(X)$ is discretely absolutely star-Lindelöf. \item {(3)} If $X$ is a normal discretely star-Lindelöf space with $e(X)< \omega _1$, then $A(X)$ is discretely absolutely star-Lindelöf.