# zbMATH — the first resource for mathematics

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.)
Full Text: