Geodesic
On spaces with point-countable $k$-systems
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 4, pp. 749-765.

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This paper deals with the behavior of $M$-spaces, countably bi-quasi-$k$-spaces and singly bi-quasi-$k$-spaces with point-countable $k$-systems. For example, we show that every $M$-space with a point-countable $k$-system is locally compact paracompact, and every separable singly bi-quasi-$k$-space with a point-countable $k$-system has a countable $k$-system. Also, we consider equivalent relations among spaces entried in Table 1 in Michael's paper [15] when the spaces have point-countable $k$-systems.
Classification : 54D55, 54E18, 54E35
Keywords: countably-bi-quasi-$k$-space; point-countable $k$-system; local compactness; \newline metrizability 
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Yoshioka, Iwao. On spaces with point-countable $k$-systems. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 4, pp. 749-765. http://geodesic.mathdoc.fr/item/CMUC_2004__45_4_a15/