Special refinements and their applications on products
The concept of a special refinement for the product of two spaces is introduced. The following are shown: (1) If \(X\) is \(HCP\)-netted and \(X \times Y\) is normal or countably paracompact, then every open cover of \(X \times Y\) has a special refinement. (2) If \(X\) is paracompact, \(Y\) is collectionwise normal (resp. collectionwise normal and countably paracompact), and every open cover (resp. every directed open cover) of \(X \times Y\) has a special refinement. Then a closed subset of \(X \times Y\) is \(P\)-embedded if and only if it is \(C\)-embedded. (3) Let \(X\) and \(Y\) be paracompact (resp. metacompact, subparacompact, submetacompact) spaces. If an open cover of \(X \times Y\) has a special refinement, then it has a \(\sigma\)-locally finite open refinement (resp. \(\sigma\)-point-finite open refinement, \(\sigma\)-locally finite closed refinement, \(\theta\)-sequence of open refinements).
scientific article; zbMATH DE number 1462706