Decompositions of spaces determined by compact subsets

Abstract:

Let $X$ be a $k’$-space, and let $\mathcal {F}$ be a closed cover of (locally) compact subsets of $X$. Then $X$ is decomposed into a closed discrete subset and a locally compact subset if $X$ is dominated by $\mathcal {F}$, or $X$ has the weak topology with respect to a point-countable cover $\mathcal {F}$. Here, a cover of a space is point-countable if every point is in at most countably many elements of the cover.