Communications of the
Korean Mathematical Society CKMS

In this paper, we introduce a new property $(\ast)$ of a topological space and prove that if $X$ satisfies one of the following conditions (1) and (2), then compactness, countable compactness and sequential compactness are equivalent in $X$; (1) Each countably compact subspace of $X$ with $(\ast)$ is a sequential or AP space. (2) $X$ is a sequential or AP space with $(\ast)$.

Keywords: Fr\'echet-Urysohn, sequential, AP, WAP, countable tightness, weakly discretely generated, compact, countably compact, sequentially compact and property $(\ast)$

MSC numbers: 54A20, 54B05, 54D30, 54D55, 54E25