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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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