Compacta and their function spaces

We consider some properties of the compact Hausdorff space X   that are defined in terms of subspaces of C(X,R)C(X,R) or C(X,C)C(X,C). In some (but not all) cases, we show that these are equivalent to more standard set-theoretic properties of X, expressed using cardinal functions such as weight, cellularity, etc.