Paracompactness with respect to an ideal
Summary: An ideal on a set \(X\) is a nonempty collection of subsets of \(X\) closed under the operations of subset and finite union. Given a topological space \(X\) and an ideal \({\mathcal I}\) of subsets of \(X\), \(X\) is defined to be \({\mathcal I}\)-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all of \(X\) except for a set in \({\mathcal I}\). Basic results are investigated, particularly with regard to the \({\mathcal I}\)-paracompactness of two associated topologies generated by sets of the form \(U-I\) where \(U\) is open and \(I\in {\mathcal I}\) and \(\bigcup \{U\mid U\) is open and \(U- A\in {\mathcal I}\), for some open set \(A\}\). Preservation of \({\mathcal I}\)-paracompactness by functions, subsets, and products is investigated. Important special cases of \({\mathcal I}\)-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of \textit{M. K. Singal} and \textit{S. P. Arya} [Math. Ann. 181, 119-133 (1969; Zbl 0162.26404)].
scientific article; zbMATH DE number 1054247