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Absolute weak \(C\)-embedding in Hausdorff spaces. (English) Zbl 1025.54010
A subspace \(Y\) of a topological space \(X\) is weakly \(C\)-embedded in \(X\) if every real-valued continuous function on \(Y\) can be extended to a real-valued function on \(X\) which is continuous at each point of \(Y\). In [Sci. Math. J. 55, 153-201 (2002; Zbl 0994.54024)], A. V. Arhangel’skiĭ has asked the question of when a Hausdorff space \(Y\) can be weakly \(C\)-embedded in every larger Hausdorff space \(X\). The author gives the following answers: Let \(Y\) be a Hausdorff space [resp., regular \(T_1\)-space]. Then \(Y\) can be weakly \(C\)-embedded in every larger Hausdorff space [resp., every larger regular \(T_1\)-space] \(X\) if and only if either \(Y\) is compact or every real-valued continuous function on \(Y\) is constant [resp., either \(Y\) is Lindelöf or of every two disjoint zero-sets of \(Y\) at least one is compact]. The author has also proved corresponding results for weak \(P\)-embedding in the sense of [T. Hoshina and the author, Topology Appl. 125, 233-247 (2002; Zbl 1013.54006)].

MSC:
54C25 Embedding
54C20 Extension of maps
54C45 \(C\)- and \(C^*\)-embedding
54D30 Compactness
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