On C-embedded subspaces of the Sorgenfrey plane


  • Olena Karlova Chernivtsi National University




$C^*$-embedded, $C$-embedded, the Sorgenfrey plane.


We show that for a subspace $E\subseteq\{(x,-x):x\in\mathbb R\}$ of the Sorgenfrey plane $\mathbb S^2$ the following conditions are equivalent: (i) $E$ is $C$-embedded in $\mathbb S^2$; (ii) $E$ is $C^*$-embedded in $\mathbb S^2$; (iii) $E$ is a countable $G_\delta$-subspace of $\rr$ and (iv) $E$ is a countable functionally closed subspace of $\ss$. We also prove that $\mathbb S^2$ is not a $\delta$-normally separated space.


Download data is not yet available.

Author Biography

Olena Karlova, Chernivtsi National University

Assistant professor of the Department of Mathematical Analysis


W.Bade, Two properties of the Sorgenfrey plane, Pacif. J. Math. 51, no. 2 (1974), 349-354. (http://dx.doi.org/10.2140/pjm.1974.51.349)

R. Blair and A.Hager, Extensions of zero-sets and of real-valued functions, Math. Zeit. 136 (1974), 41-52. (http://dx.doi.org/10.1007/BF01189255)

G.Debs, Espaces héréditairement de Baire, Fund. Math. 129, no. 3 (1988), 199-206.

R. Engelking, General Topology. Revised and completed edition. Heldermann Verlag, Berlin (1989).

L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton (1960). (http://dx.doi.org/10.1007/978-1-4615-7819-2)

R. Heath and E. Michael, A property of the Sorgenfrey line, Comp. Math. 23, no. 2 (1971), 185-188.

T. Hoshina and K. Yamazaki, Weak C-embedding and P-embedding, and product spaces, Topology Appl. 125 (2002), 233-247. (http://dx.doi.org/10.1016/S0166-8641(01)00275-9)

O.Kalenda and J.Spurny, Extending Baire-one functions on topological spaces, Topology Appl. 149 (2005), 195-216. (http://dx.doi.org/10.1016/j.topol.2004.09.007)

O. Karlova, On $alpha$-embedded sets and extension of mappings, Comment. Math. Univ. Carolin. 54, no. 3 (2013), 377-396.

J.Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math. Soc. 148 (1970), 265-272. (http://dx.doi.org/10.1090/S0002-9947-1970-0259856-3)

H. Ohta, Extension properties and the Niemytzki plane, Appl. Gen. Topol. 1, no. 1 (2000), 45-60.

H. Ohta and K. Yamazaki, Extension problems of real-valued continuous functions, in: ''Open problems in topology II'', E.Pearl (ed.), Elsevier, 2007, 35-45. (http://dx.doi.org/10.1016/B978-044452208-5/50005-3)

J.Saint-Raymond, Jeux topologiques et espaces de Namioka, Proc. Amer. Math. Soc. 87, no. 3 (1983), 409-504. (http://dx.doi.org/10.1090/S0002-9939-1983-0684646-1)

W.Sierpinski, Sur une propriete topologique des ensembles denombrables denses en soi, Fund. Math. 1 (1920), 11-16.

Y.Tanaka, On closedness of $C$- and $C^*$-embeddings, Pacif. J. Math. 68, no. 1 (1977), 283-292. (http://dx.doi.org/10.2140/pjm.1977.68.283)

J. Terasawa, On the zero-dimensionality of some non-normal product spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 11 (1972), 167-174.




How to Cite

O. Karlova, “On C-embedded subspaces of the Sorgenfrey plane”, Appl. Gen. Topol., vol. 16, no. 1, pp. 65–74, Feb. 2015.