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- W1987187813 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 Marczewskiâs 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." @default.
- W1987187813 created "2016-06-24" @default.
- W1987187813 creator A5085422980 @default.
- W1987187813 date "2009-09-01" @default.
- W1987187813 modified "2023-09-27" @default.
- W1987187813 title "Menger subsets of the Sorgenfrey line" @default.
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- W1987187813 doi "https://doi.org/10.1090/s0002-9939-09-09887-6" @default.
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