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Proceedings of the American Mathematical Society

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The pseudocompact extension $ \alpha X$


Authors: C. E. Aull and J. O. Sawyer
Journal: Proc. Amer. Math. Soc. 102 (1988), 1057-1064
MSC: Primary 54C45; Secondary 54D30
MathSciNet review: 934890
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Abstract: For any Tychonoff space we define $ \alpha X = (\beta X - vX) \cup X = \beta X - (vX - X)$. We show that $ \alpha X$ is the smallest pseudocompactification $ Y$ of $ X$ contained is $ \beta X$ such that every free hyperreal $ z$-ultrafilter on $ X$ converges in $ Y$ and is the largest pseudocompactification $ Y$ of $ X$ contained in $ \beta X$ such that every point in $ Y - X$ is contained in a zero set of $ Y$ which does not intersect $ X$. A space $ S$ is defined to be $ \alpha $-embedded in a space $ X$ if $ \alpha S \subset \beta X$. Properties of $ \alpha $-embeddings and its relation to $ v$-embeddings of Blair $ {C^*}$-embeddings, $ C$-embeddings, and well-embeddings are investigated.

For instance, if $ S$ is $ \alpha $-embedded and dense in $ X,S$ is fully well-embedded (for $ P,R \subset X$, where $ S \subset P \subset R \subset X$, $ P$ is well-embedded in $ R$) in $ X$ iff $ \alpha X - \alpha S = X - S$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0934890-5
Article copyright: © Copyright 1988 American Mathematical Society