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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Wallman-type compactifications


Author: Charles M. Biles
Journal: Proc. Amer. Math. Soc. 25 (1970), 363-368
MSC: Primary 54.53
DOI: https://doi.org/10.1090/S0002-9939-1970-0263029-3
MathSciNet review: 0263029
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Abstract: All spaces in this paper are Tychonoff. A Wallman base on a space $ X$ is a normal separating ring of closed subsets of $ X$ (see Steiner, Duke Math. J. 35 (1968), 269-276). Let $ T$ be a compact space and $ \mathcal{L}$ a Wallman base on $ T$. For $ X \subset T$, define $ {\mathcal{L}_X} = \{ A \cap X\vert A \in \mathcal{L}\} $.

Theorem 1. If $ X$ is a dense subspace of $ T$, then $ T = w{\mathcal{L}_X}$ iff $ {\operatorname{cl} _T}A \cap {\operatorname{cl} _T}B = \emptyset $ whenever $ A,B \in {\mathcal{L}_X}$ and $ A \cap B = \emptyset $.

Theorem 2. $ T = w{\mathcal{L}_X}$ for each dense $ X \subset T$ iff $ T = w{\mathcal{L}_Y}$ for each dense $ Y \subset T$ where $ T - Y \in \mathcal{L}$.

From these theorems we show that every compact $ F$-space and every compact orderable space is a Wallman compactification of each of its dense subspaces.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0263029-3
Keywords: Tychonoff space, compact space, Hausdorff compactification, Wallman base, Wallman compactification, $ z$-compactification, dense subspace, compact $ F$-space, compact orderable space
Article copyright: © Copyright 1970 American Mathematical Society