$\kappa$-spaces

Abstract

We say that a Tychonoff space $X$ is a $\kappa$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $\kappa$-spaces is strictly between the class of Dieudonn\'{e} complete spaces and the class of $\mu$-spaces. We show that the class of $\kappa$-spaces has nice stability properties, that allows us to define the $\kappa$-completion $\kappa X$ of $X$ as the smallest $\kappa$-space in the Stone--\v{C}ech compactification $\beta X$ of $X$ containing $X$. For a point $z\in\beta X$, we show that (1) if $z\in\upsilon X$, then the Dirac measure $\delta_z$ at $z$ is bounded on each compact subset of $C_p(X)$, (2) $z\in \kappa X$ iff $\delta_z$ is continuous on each compact subset of $C_p(X)$ iff $\delta_z$ is continuous on each compact subset of $C_p^b(X)$, (3) $z\in\upsilon X$ iff $\delta_z$ is bounded on each compact subset of $C_p^b(X)$. It is proved that $\kappa X$ is the largest subspace $Y$ of $\beta X$ containing $X$ for which $C_p(Y)$ and $C_p(X)$ have the same compact subsets, this result essentially generalizes a known result of R.~Haydon.