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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
Citations:
Zbl 0238.54009
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