Beschreibung:
It is a familiar fact that | C ( X ) | ≦ 2 δ X |C(X)| \leqq {2^{\delta X}} , where | C ( X ) | |C(X)| is the cardinal number of the set of real-valued continuous functions on the infinite topological space X X , and δ X \delta X is the least cardinal of a dense subset of X X . While for metrizable spaces equality obtains, for some familiar spaces—e.g., the one-point compactification of the discrete space of cardinal 2 ℵ 0 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 ) | |C(X)| which does not involve a cardinal property of X X as an exponent, and therefore we consider exponential combinations of certain natural cardinal numbers associated with X X . Among the numbers are w X wX , the least cardinal of an open basis, and w c X wcX , the least m \mathfrak {m} for which each open cover of X X has a subfamily with m \mathfrak {m} or fewer elements whose union is dense. We show that | C ( X ) | ≦ ( w X ) w c X |C(X)| \leqq {(wX)^{wcX}} , and that this estimate is best possible among the numbers in question. (In particular, ( w X ) w c X ≦ 2 δ X {(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 X for which | C ( X ) | > ( w X ) w c X |C(X)| > {(wX)^{wcX}} .