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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)

##### MSC:
 54E18 $$p$$-spaces, $$M$$-spaces, $$\sigma$$-spaces, etc. 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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