Relative topological properties and relative topological spaces
This is a highly interesting survey article on so-called relative topological properties. It has to do with the question of how a given subspace \(Y\) of a space \(X\) is ``located'' in \(X\). The basic idea is as follows: with each topological property \(\mathbb{P}\) one can associate a relative version of it formulated in terms of location of \(Y\) in \(X\) in such a way that when \(Y\) coincides with \(X\) then this relative property coincides with \(\mathbb{P}\). In the first part of this paper relative separation axioms are considered. Most results revolve around the situation when a subspace \(Y\) is normal (resp. strongly normal) in \(X\). \(Y \subseteq X\) is normal in \(X\) if for each pair \(A,B\) of closed disjoint subsets of \(X\) there are disjoint open subsets \(U\) and \(V\) in \(X\) such that \(A \cap Y \subseteq U\) and \(B \cap Y \subseteq V\). \(Y \subseteq X\) is strongly normal in \(Y\) if for each pair \(A,B\) of closed in \(Y\) disjoint subsets of \(Y\) there are open disjoint subsets \(U\) and \(V\) in \(X\) such that \(A \subseteq U\) and \(B \subseteq V\). In the second part of the paper it is discussed how several relative compactness type properties influence relative separation properties. Typical results mentioned there are: (i) If \(Y \) is Lindelöf in a regular space \(X\) then \(Y\) is paracompact in \(X\), and (ii) if \(Y\) is paracompact in \(X\), and \(X\) is regular, then \(Y\) is normal in \(X\). Throughout the paper a large number of still open problems are posed, thus setting the direction for possible future research.
scientific article; zbMATH DE number 896606