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Menger subsets of the Sorgenfrey line

Author: Masami Sakai
Journal: Proc. Amer. Math. Soc. 137 (2009), 3129-3138
MSC (2000): Primary 03E15; Secondary 54D20, 54H05
Published electronically: March 24, 2009
MathSciNet review: 2506472
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Abstract: 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.

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Additional Information

Masami Sakai
Affiliation: Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan

Keywords: Sorgenfrey line, Menger property, Hurewicz property, property ($s^0$), totally imperfect, universally meager, $\lambda $-set, totally paracompact, totally metacompact, GO-space
Received by editor(s): November 13, 2008
Received by editor(s) in revised form: January 10, 2009
Published electronically: March 24, 2009
Additional Notes: This work was supported by KAKENHI (No. 19540151)
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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