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On spaces with point-countable $$k$$-systems. (English) Zbl 1099.54025
Summary: 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 equivalence relations among the spaces in Table 1 in E. A. Michael [General Topology Appl. 2, 91-138 (1972; Zbl 0238.54009)] when the spaces have point-countable $$k$$-systems.
##### MSC:
 54D55 Sequential spaces 54E18 $$p$$-spaces, $$M$$-spaces, $$\sigma$$-spaces, etc. 54E35 Metric spaces, metrizability
Zbl 0238.54009
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