Local compactness with respect to an ideal.

*(English)*Zbl 0767.54019Summary: 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.

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 |