Some generalizations of the concept of a \(p\)-space. (English) Zbl 1229.54036

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.
Reviewer: K. P. Hart (Delft)


54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
Full Text: DOI


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