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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
DOI: https://doi.org/10.1090/S0002-9947-1970-0263016-X
MathSciNet review: 0263016
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Abstract: It is a familiar fact that $|C(X)| \leqq {2^{\delta X}}$, where $|C(X)|$ 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 $|C(X)|$ 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 $|C(X)| \leqq {(wX)^{wcX}}$, 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 $|C(X)| < {(wX)^{wcX}}$.


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Keywords: Real-valued continuous functions, weight, density character, Stone-&#268;ech compactification, Marczewski Theorem, &#352;anin Theorem, continuum hypothesis
Article copyright: © Copyright 1970 American Mathematical Society