Menger subsets of the Sorgenfrey line

Abstract

Abstract. 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 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 ⊂ X \ Q, then there is an embedding e : X → R such that e(X) 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.

Topological Games and Alster Spaces

“…Pick a Sierpiński set S ⊆ R and endow it with the Sorgenfrey line topology. By Corollary 3.6 of [28], S is Menger. Since S does not have measure zero, it cannot be Rothberger [27]; thus, as every compact subset of the Sorgenfrey line is countable, it follows from Corollary 3.9 that S does not satisfy S 1 (K, O).…”

Menger-type covering properties of topological spaces

“…Let S be the Sorgenfrey line and let R be the set of reals. If i : S → R is the identity map and X ⊂ R then by X S we denote i −1 (X) (see [20]). Lelek showed in [16] that for every Lusin set L in R, L S is Menger and, therefore, almost Menger, but if L satisfies that (L × L) ∩ {(x, y) : x + y = 0} is uncountable, then L S × L S is not Menger and since S × S is regular and every subspace of a regular space is regular, we have that the square of L S is regular and not Menger.…”

Base-base paracompactness and subsets of the Sorgenfrey line

“…Such is any Lusin set or, under MA, any uncountable set of cardinality less than continuum. These sets are Hurewicz [9], [13], and hence totally paracompact [6], see also [2], [12], [14], [15], [25]. G. Gruenhage gave a direct proof for Lusin subspaces that the base of all half-open intervals contains no coarse base.…”