Menger subsets of the Sorgenfrey line

A space $X$ is said to have the Menger property if for every sequence $\{\mathcal{U}_n:n \in \omega \}$ of open covers of $X$, there are finite subfamilies $\mathcal{V}_n \subset \mathcal{U}_n$ ( $n \in \omega$) such that $\bigcup_{n \in \omega}\mathcal{V}_n$ is a cover of $X$. Let $i:\mathbb{S} \to \mathbb{R}$ be the identity map from the Sorgenfrey line onto the real line and let $X_\mathbb{S}=i^{-1}(X)$ for $X \subset \mathbb{R}$. Lelek noted in 1964 that for every Lusin set $L$ in $\mathbb{R}$, $L_\mathbb{S}$ has the Menger property. In this paper we further investigate Menger subsets of the Sorgenfrey line. Among other things, we show: (1) If $X_\mathbb{S}$ has the Menger property, then $X$ has Marczewski's property ($s^0$). (2) Let $X$ be a zero-dimensional separable metric space. If $X$ has a countable subset $Q$ satisfying that $X \setminus A$ has the Menger property for every countable set $A \subset X \setminus Q$, then there is an embedding $e:X \to \mathbb{R}$ such that $e(X)_\mathbb{S}$ has the Menger property. (3) For a Lindelöf subspace of a real GO-space (for instance the Sorgenfrey line), total paracompactness, total metacompactness and the Menger property are equivalent.