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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Estimates for the number of real-valued continuous functions

Authors: W. W. Comfort and Anthony W. Hager
Journal: Trans. Amer. Math. Soc. 150 (1970), 619-631
MSC: Primary 54.28
MathSciNet review: 0263016
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Abstract: It is a familiar fact that $ \vert C(X)\vert \leqq {2^{\delta X}}$, where $ \vert C(X)\vert$ is the cardinal number of the set of real-valued continuous functions on the infinite topological space $ X$, and $ \delta X$ is the least cardinal of a dense subset of $ X$. While for metrizable spaces equality obtains, for some familiar spaces--e.g., the one-point compactification of the discrete space of cardinal $ 2\aleph 0$--the inequality can be strict, and the problem of more delicate estimates arises. It is hard to conceive of a general upper bound for $ \vert C(X)\vert$ which does not involve a cardinal property of $ X$ as an exponent, and therefore we consider exponential combinations of certain natural cardinal numbers associated with $ X$. Among the numbers are $ wX$, the least cardinal of an open basis, and $ wcX$, the least $ \mathfrak{m}$ for which each open cover of $ X$ has a subfamily with $ \mathfrak{m}$ or fewer elements whose union is dense. We show that $ \vert C(X)\vert \leqq {(wX)^{wcX}}$, and that this estimate is best possible among the numbers in question. (In particular, $ {(wX)^{wcX}} \leqq {2^{\delta X}}$ always holds.) In fact, it is only with the use of a version of the generalized continuum hypothesis that we succeed in finding an $ X$ for which $ \vert C(X)\vert < {(wX)^{wcX}}$.

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Keywords: Real-valued continuous functions, weight, density character, Stone-Čech compactification, Marczewski Theorem, Šanin Theorem, continuum hypothesis
Article copyright: © Copyright 1970 American Mathematical Society

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