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Proceedings of the American Mathematical Society

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

 

 

Pseudocompactness properties


Author: Samuel Broverman
Journal: Proc. Amer. Math. Soc. 59 (1976), 175-178
MSC: Primary 54D60
DOI: https://doi.org/10.1090/S0002-9939-1976-0413050-2
MathSciNet review: 0413050
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Abstract: A topological extension property is a class of Tychonoff spaces $ \mathcal{P}$ which is closed hereditary, closed under formation of topological products and contains all compact spaces. If $ X$ is Tychonoff and $ \mathcal{P}$ is an extension property, there is a space $ \mathcal{P}X$ such that $ X \subseteq \mathcal{P}X \subseteq \beta X,\;\mathcal{P}X \in \mathcal{P}$ and if $ f:X \to Y$ where $ Y \in \mathcal{P}$ then $ f$ admits a continuous extension to $ \mathcal{P}X$. A space $ X$ is called $ \mathcal{P}$-pseudocompact if $ \mathcal{P}X = \beta X$. In this note it is shown that if $ \mathcal{P}$ is an extension property which contains the real line (e.g., the class of realcompact spaces), $ X$ is $ \mathcal{P}$-pseudocompact and $ Y$ is compact, then $ X \times Y$ is $ \mathcal{P}$-pseudocompact. An example is given of an extension property $ \mathcal{P}$, a $ \mathcal{P}$-pseudocompact space $ X$ and a compact space $ Y$ such that $ X \times Y$ is not $ \mathcal{P}$-pseudocompact.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0413050-2
Keywords: Stone-Čech compactification, extension property, realcompact, pseudocompact
Article copyright: © Copyright 1976 American Mathematical Society