Geodesic
Spaces with large relative extent
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 387-394 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper, we prove the following statements: (1) For every regular uncountable cardinal $\kappa $, there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e(Y,X) \ge \kappa $, but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf in $X$.
In this paper, we prove the following statements: (1) For every regular uncountable cardinal $\kappa $, there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e(Y,X) \ge \kappa $, but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf in $X$.
Classification : 54D15, 54D20
Keywords: relative topological property; Lindelöf; star-Lindelöf; relative extent; relative property (a) 
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Song, Yan-Kui. Spaces with large relative extent. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 387-394. http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a28/

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