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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)].

54C25 Embedding
54C20 Extension of maps
54C45 \(C\)- and \(C^*\)-embedding
54D30 Compactness
Full Text: DOI
[1] Alò, R.A.; Shapiro, H.L., Normal topological spaces, (1974), Cambridge University Press Cambridge · Zbl 0282.54005
[2] Arhangel’skiı̆, A.V., Relative topological properties and relative topological spaces, Topology appl., 70, 87-99, (1996) · Zbl 0848.54016
[3] Arhangel’skiı̆, A.V., From classic topological invariants to relative topological properties, Sci. math. japon., 55, 153-201, (2002) · Zbl 0994.54024
[4] Arhangel’skiı̆, A.V.; Tartir, J., A characterization of compactness by relative separation property, Questions answers gen. topology, 14, 49-52, (1996) · Zbl 0851.54001
[5] Bella, A.; Yaschenko, I.V., Lindelöf property and absolute embeddings, Proc. amer. math. soc., 127, 907-913, (1999) · Zbl 0907.54003
[6] Blair, R.L., On υ-embedded sets in topological spaces, (), 46-79
[7] Blair, R.L.; Hager, A.W., Extensions of zero-sets and of real-valued functions, Math. Z., 136, 41-52, (1974) · Zbl 0264.54011
[8] Doss, R., On uniform spaces with a unique structure, Amer. J. math., 71, 19-23, (1949) · Zbl 0032.12202
[9] Engelking, R., General topology, (1989), Heldermann Berlin · Zbl 0684.54001
[10] Gartside, P.M.; Glyn, A., Relative separation properties, Topology appl., 122, 625-636, (2002) · Zbl 1005.54024
[11] Gillman, L.; Jerison, M., Rings of continuous functions, (1960), Van Nostrand Princeton, NJ · Zbl 0093.30001
[12] Hager, A.W.; Johnson, D.G., A note on certain subalgebras of C(X), Canad. J. math., 20, 389-393, (1968) · Zbl 0162.26702
[13] Hewitt, E., A note on extensions of continuous functions, An. acad. brasil. ci., 21, 175-179, (1949)
[14] Hoshina, T.; Yamazaki, K., Weak C-embedding and P-embedding, and product spaces, Topology appl., 125, 233-247, (2002) · Zbl 1013.54006
[15] Matveev, M.V.; Pavlov, O.I.; Tartir, J.K., On relatively normal spaces, relatively regular spaces, and on relative property (a), Topology appl., 93, 121-129, (1999) · Zbl 0951.54017
[16] Smirnov, Y., Mappings of systems of open sets, Mat. sb., 31, 152-166, (1952), (in Russian) · Zbl 0047.16102
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