On $C$-embedded subspaces of the Sorgenfrey plane

Abstract

We prove that every $C^{\ast }$-embedded subset of $ß$ is a hereditarily Baire subspace of ${\mathbb{R}}^2$. We also 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 $C^{\ast }$-embedded in ${\mathbb{S}}^2$; (iii) $E$ is a countable $G_{\delta }$-subspace of ${\mathbb{R}}^2$ and (iv) $E$ is a countable functionally closed subspace of ${\mathbb{S}}^2$.

arXiv commentin Applied General Topology 2015