Some generalizations of the concept of a \(p\)-space
The authors define generalizations of the notion of a \(p\)-space. We say that a subset~\(X\) of a space~\(Z\) is \(s\)-embedded if there is a countable family~\(\mathcal S\) of open subsets such that \(X\)~can be written as the union of a family of subsets, each of which is the intersection of a subfamily of~\(\mathcal S\). An \(s\)-space is one which is \(s\)-embedded in one of its compactifications. A partial pluming of~\(X\) in~\(Z\) is a sequence \(\langle\gamma_n:n\in\omega\rangle\) of families of open sets such that whenever \(x\in X\) and \(z\in Z\setminus X\) there is an~\(n\) such that \(x\in\bigcup\gamma_n\) and \(z\notin\text{St}(x,\gamma_n)\); the sequence is a pluming if \(X\subset\bigcup\gamma_n\) for all~\(n\). If \(X\)~has a (partial) pluming in~\(Z\) then it is said to be \(p\)-embedded (\(p^*\)-embedded) in~\(Z\); a \(p\)-space (\(p^*\)-space) is one that is \(p\)-embedded (\(p^*\)-embedded) in some compactification. Besides establishing the basic properties of and the relations between these notions the authors also apply them to sufficient conditions for and characterizations of being a \(p\)-space or metrizable.
scientific article; zbMATH DE number 5935885
scientific article