  In this paper we find sufficient and necessary conditions under which vector
lattice $C(X)$ and its sublattices $C_b(X)$, $C_0(X)$ and $C_c(X)$ have the
countable sup property. It turns out that the countable sup property is tightly
connected to the countable chain condition of the underlying topological space
$X$. We also consider the countable sup property of $C(X\times Y)$. Even when
both $C(X)$ and $C(Y)$ have the countable sup property it is possible that
$C(X\times Y)$ fails to have it. For this construction one needs to assume the
continuum hypothesis. In general, we present a positive result in this
direction and also address the question when $C(\prod_{\lambda\in\Lambda}
X_\lambda)$ has the countable sup property. Our results can be understood as
vector lattice theoretical versions of results regarding products of spaces
satisfying the countable chain condition. We also present new results for
general vector lattices that are of an independent interest.
