The countable sup property for lattices of continuous functions 

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.