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Proceedings of the American Mathematical Society

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

 
 

 

Products of uncountably many $ k$-spaces


Author: N. Noble
Journal: Proc. Amer. Math. Soc. 31 (1972), 609-612
DOI: https://doi.org/10.1090/S0002-9939-1972-0287503-0
MathSciNet review: 0287503
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Abstract: It is shown that if a product of nonempty spaces is a $ k$-space then for each infinite cardinal $ \mathfrak{n}$ some product of all but $ \mathfrak{n}$ of the factors has each $ \mathfrak{n}$-fold subproduct $ \mathfrak{n} - {\aleph _0}$-compact (each $ \mathfrak{n}$-fold open cover has a finite subcover). An example is given, for each regular $ \mathfrak{n}$, of a space $ X$ which is not $ \mathfrak{n} - {\aleph _0}$-compact (so $ {X^{{\mathfrak{n}^ + }}}$ is not a $ k$-space) for which $ {X^\mathfrak{n}}$ is a $ k$-space.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0287503-0
Keywords: Product spaces, $ k$-spaces, spaces with weak topologies
Article copyright: © Copyright 1972 American Mathematical Society

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