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Proceedings of the American Mathematical Society

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

 

 

For any $ X$, the product $ X\times Y$ has remote points for some $ Y$


Author: Thomas J. Peters
Journal: Proc. Amer. Math. Soc. 95 (1985), 641-648
MSC: Primary 54D40; Secondary 54A25, 54B10, 54B25, 54D35, 54G20
MathSciNet review: 810178
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Abstract: Any space with a $ \sigma $-locally finite $ \pi $-base will be called a $ \sigma - \pi $ space. The work of Chae and Smith can be extended to show that every nonpseudocompact $ \sigma - \pi $ space has remote points.$ ^{2}$ Sufficient conditions for a product to be a $ \sigma - \pi $ space are developed. It is shown that, for each space, if $ \alpha $ is a cardinal with the discrete topology, where $ \alpha $ is not less than $ \pi $-weight of $ X$, then $ X \times {\alpha ^\omega }$ has remote points. Cardinal function criteria are developed for the existence of $ \sigma - \pi $ spaces. An example is given of a pathological product which is a $ \sigma - \pi $ space even though none of its finite partial products is a $ \sigma - \pi $ space.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0810178-5
Keywords: $ \sigma - \pi $ space, products, $ G$-space, $ \sigma $-locally finite $ \pi $-base, remote point, Stone-Čech compactification, remainder, homogeneity, pseudo-$ \gamma $-compact
Article copyright: © Copyright 1985 American Mathematical Society