# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.