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Menger subsets of the Sorgenfrey line

A space X is said to have the Menger property if for every sequence {u n : n ∈ ω} of open covers of X, there are finite subfamilies V n ⊂ U n (n, ∈ ω) such that U n∈ω V n is a cover of X. Let i: S → R be the identity map from the Sorgenfrey line onto the real line and let X S = i -1 (X) for X ⊂ ℝ. Lelek noted in 1964 that for every Lusin set L in ℝ, L 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 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 \ A has the Menger property for every countable set A C X \ Q, then there is an embedding e: X → R such that e(X) S has the Menger property. (3) For a Lindelof subspace of a real GO-space (for instance the Sorgenfrey line), total paracompactness, total metacompactness and the Menger property are equivalent.