Estimates for the number of real-valued continuous functions

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 $|C(X)| < {(wX)^{wcX}}$.