Fund. Math. 200 (2008), no. 2, 185–199.1730-6329 Summary: “Let Cp(X) be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) Cp(X) is Scott-domain representable; (b) Cp(X) is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T1-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that Cp(X) is subcompact if and only if X is discrete.