A filter property of submetacompactness and its application to products
From the author's introduction and abstract: ``If \({\mathcal V}\) is a collection of subsets of a set X, let ord(x,\({\mathcal V})\) denote the cardinality of \(\{\) \(V\in {\mathcal V}:\) \(x\in V\}\). A space is metacompact if every open cover of X has a point-finite open refinement and is submetacompact if every open cover \({\mathcal U}\) has a sequence (\({\mathcal V}_ n)\) of open refinements such that for each \(x\in X\), ord(x,\({\mathcal V}_ n)<\omega\) for some \(n\in \omega...''\). ``We show that there is a filter on \(\omega\) such that for any submetacompact X and any open cover \({\mathcal U}\) of X, there is a sequence \(\{\) \({\mathcal V}_ n:\) \(n\in \omega \}\) of open refinements of \({\mathcal U}\) such that \(\{\) n: ord(x,\({\mathcal V}_ n)<\omega \}\) is in the filter for every \(x\in X\). We apply this result to submetacompactness of product spaces showing, e.g., that if X has a \(\sigma\)-closure preserving cover by compact sets, then \(X\times Y\) is submetacompact for every submetacompact space Y.''
scientific article; zbMATH DE number 4192609
scientific article