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Transactions of the American Mathematical Society

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Factoring functions on Cartesian products


Authors: N. Noble and Milton Ulmer
Journal: Trans. Amer. Math. Soc. 163 (1972), 329-339
MSC: Primary 54.25
DOI: https://doi.org/10.1090/S0002-9947-1972-0288721-2
MathSciNet review: 0288721
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Abstract: A function on a product space is said to depend on countably many coordinates if it can be written as a function defined on some countable subproduct composed with the projection onto that subproduct. It is shown, for $ X$ a completely regular Hausdorff space having uncountably many nontrivial factors, that each continuous real-valued function on $ X$ depends on countably many coordinates if and only if $ X$ is pseudo- $ {\aleph _1}$-compact. It is also shown that a product space is pseudo- $ {\aleph _1}$-compact if and only if each of its finite subproducts is. (This fact derives from a more general theorem which also shows, for example, that a product satisfies the countable chain condition if and only if each of its finite subproducts does.) All of these results are generalized in various ways.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0288721-2
Keywords: Functions depending on countably many coordinates, pseudo- $ \mathfrak{n}$-compact spaces, countable chain condition, realcompactifications of infinite products, infinite products
Article copyright: © Copyright 1972 American Mathematical Society

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