κ-spaces

We say that a Tychonoff space X is a κ-space if it is homeomorphic to a closed subspace of Cp(Y) for some locally compact space Y. The class of κ-spaces is strictly between the class of Dieudonné complete spaces and the class of μ-spaces. We show that the class of κ-spaces has nice stability properties, that allows us to define the κ-completion κX of X as the smallest κ-space in the Stone–Čech compactification βX of X containing X. For a point z∈βX, we show that (1) if z∈υX, then the evaluation function δz at z is bounded on each compact subset of Cp(X), (2) z∈κX iff δz is continuous on each compact subset of Cp(X) iff δz is continuous on each compact subset of Cpb(X), (3) z∈υX iff δz is bounded on each compact subset of Cpb(X). It is proved that κX is the largest subspace Y of βX containing X for which Cp(Y) and Cp(X) have the same compact subsets, this result essentially generalizes a known result of R. Haydon.