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Local compactness with respect to an ideal. (English) Zbl 0767.54019
Summary: Given a nonempty set $$X$$, an ideal $$\mathcal I$$ on $$X$$ is a collection of subsets of $$X$$ closed under finite union and subset operations. In [“Compactness modulo an ideal”, Sov. Math., Dokl. 13, No. 1, 193-197 (1972); translation from Dokl. Akad. Nauk SSSR 202, 761-764 (1972; Zbl 0254.54023)] D. V. Rančin studied a generalization of compactness ($$\mathcal I$$-compactness) which requires that open covers of a space have a finite subcollection which covers all the space except for a set in the ideal.
We define a space to be locally $$\mathcal I$$-compact if every point in the space has an $$\mathcal I$$-compact neighborhood. Basic results concerning locally $$\mathcal I$$-compact spaces are given relating to subspaces, preservation by functions, and products. Classical results concerning locally compact spaces are obtained by letting $${\mathcal I}=\{\emptyset\}$$, and certain results for locally $$H$$-closed spaces are obtained by letting $$\mathcal I$$ be the ideal of nowhere dense sets. Locally $$H$$-closed spaces are characterized in the category of Hausdorff spaces as being the locally nowhere-dense-compact spaces.

##### MSC:
 54D30 Compactness 54D45 Local compactness, $$\sigma$$-compactness 54D25 “$$P$$-minimal” and “$$P$$-closed” spaces