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${\rm KC}$-space; strongly ${\rm KC}$-space; ${\rm FDS}$-property; maximal (countably) compact
In this article we introduce the notion of strongly ${\rm KC}$-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space $(X, \tau )$ is maximal countably compact if and only if it is minimal strongly ${\rm KC}$, and apply this result to study some properties of minimal strongly ${\rm KC}$-spaces, some of which are not possessed by minimal ${\rm KC}$-spaces. We also give a positive answer to a question proposed by O. T. Alas and R. G. Wilson, who asked whether every countably compact ${\rm KC}$-space of cardinality less than $c$ has the ${\rm FDS }$-property. Using this we obtain a characterization of Katětov strongly ${\rm KC}$-spaces and finally, we generalize one result of Alas and Wilson on Katětov-${\rm KC}$ spaces.
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