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Spaces determined by point-countable covers. (English) Zbl 0561.54016
This interesting paper covers a variety of topics related to point- countable covers. In addition to ”point-countable base”, there are nine conditions given which describe ways that a point-countable cover can influence the topology of a space X. For instance we list here three of the nine conditions. (1.1) X has a point-countable cover \({\mathcal P}\) such that each open \(U\subset X\) is determined by \(\{\) \(P\in {\mathcal P}:P\subset U\}\) (a space Y is said to be determined by a cover \({\mathcal Q}\) provided \(V\subset Y\) is open in Y if and only if \(V\cap Q\) is relatively open in Q for every \(Q\in {\mathcal Q})\); (1.4) X has a point-countable k-network; (1.5) X has a point countable cover \({\mathcal P}\) such that if \(X\in K\subset U\) with K compact, and U open in X, then there is a finite \({\mathcal F}\subset {\mathcal P}\) such that \(\cup {\mathcal F}\subset U\), \(x\in \cup {\mathcal F}\) and \({\mathcal F}\) covers a neighborhood of x in K. Two of the nine conditions (1.2) and \((1.2)_ p\) (not given here) were considered earlier by D. Burke and E. Michael [ibid. 64, 79-92 (1976; Zbl 0341.54022)]. The relations among the nine conditions are explored in detail. For instance, it is proved that (1.1) implies (1.4) and (1.5), and that a k-space satisfies (1.1) if and only if it is the quotient s- image of a metric space. Other topics considered are countably bi-k- spaces, separable spaces (in a regular Fréchet space, being separable is equivalent to being an \(\aleph_ 0\)-space; see the paper by E. Michael [J. Math. Mech. 15, 983-1002 (1966; Zbl 0148.167)]), and the preservation of the nine properties by the operations of subsets, quotients, countable products, and certain mappings. There are 15 examples and 10 questions raised in the paper.
Reviewer: J.E.Vaughan

MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D50 \(k\)-spaces
54E35 Metric spaces, metrizability
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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