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}}
.