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Weak \(C\)-embedding and \(P\)-embedding, and product spaces. (English) Zbl 1013.54006
\(Y\) is weakly \(P^\gamma\)-embedded in \(X\) if every continuous \(\gamma\)-pseudometric on \(Y\) extends to a pseudometric on \(X\) which is continuous on \(Y\). It is shown that if \(Y\subset X\) then \(Y\) is weakly \(P^\gamma\)-embedded in \(X\) if and only if each disjoint collection \(\{ G_\alpha\mid \alpha<\gamma\}\) of open subsets of \(Y\) for which \(\bigcup_{\alpha <\gamma}G_\alpha\) is a cozero set in \(Y\) extends to a disjoint collection of open subsets of \(X\). Other characterisations of this condition are given using product spaces.

MSC:
54C08 Weak and generalized continuity
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C20 Extension of maps
54C30 Real-valued functions in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54C25 Embedding
54B10 Product spaces in general topology
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