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.
In this paper, we study some properties of relatively strong pseudocompactness and mainly show that if a Tychonoff space $X$ and a subspace $Y$ satisfy that $Y\subset \overline {{\rm Int} Y}$ and $Y$ is strongly pseudocompact and metacompact in $X$, then $Y$ is compact in $X$. We also give an example to demonstrate that the condition $Y\subset \overline {{\rm Int} Y}$ can not be omitted.