×

On spaces with a point-countable compact \(k\)-network. (English) Zbl 0964.54023

Let \(X\) be a space, and \({\mathcal P}\) a cover of \(X\). \({\mathcal P}\) is said to be a \(k\)-network for \(X\) if \(K\subset U\) with \(K\) compact and \(U\) open, then \(K\subset\bigcup{\mathcal P}'\subset U\) for some finite subset \({\mathcal P}'\) of \({\mathcal P}\). If \(\mathcal P\) is a \(k\)-network for \(X\), then \({\mathcal P}\) is a closed (countably compact, compact) \(k\)-network if each element of \({\mathcal P}\) is closed (countably compact, compact) in \(X\). S. Lin in [Topol. Proc. 20, 185-190 (1995; Zbl 0869.54025)] proved that for a regular and \(T_1\)-space \(X\), if \(X\) has a point-countable closed \(k\)-network, then \(X\) has a point-countable countably compact \(k\)-network if and only if every first countable closed subset of \(X\) is locally compact. The following question was posed. Suppose a space \(X\) has a point-countable closed \(k\)-network. Is \(X\) a space with a point-countable compact \(k\)-network if every first countable closed subspace of \(X\) is locally compact? In the paper, the author shows that a space \(X\) is compact metrizable if and only if \(X^\omega\) has a point-countable compact \(k\)-network, and he constructs an example answering the above question in the negative by showing that there is a regular \(T_1\) countably compact space \(Y\) such that \(Y\) has a point-countable closed \(k\)-network and every first countable closed subspace of \(Y\) is compact, but \(Y\) does not have any point-countable compact \(k\)-network. Chen Huaipeng in [ibid. 24, 95-103 (1999; Zbl 0962.54025)] constructed another example giving a negative answer to the question.
Reviewer: Shou Lin (Fujian)

MSC:

54E99 Topological spaces with richer structures
54G20 Counterexamples in general topology
54D50 \(k\)-spaces
PDFBibTeX XMLCite