$\hat{I}º$-spaces

Abstract

We say that a Tychonoff space $X$ is a $\hat{I}º$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $\hat{I}º$-spaces is strictly between the class of Dieudonné complete spaces and the class of $\hat{I}¼$-spaces. We show that the class of $\hat{I}º$-spaces has nice stability properties, that allows us to define the $\hat{I}º$-completion $\hat{I}ºX$ of $X$ as the smallest $\hat{I}º$-space in the Stone--Čech compactification ${\hat{I}}^{2}X$ of $X$ containing $X$. For a point $z\in {\hat{I}}^{2}X$, we show that (1) if $z\in \upsilon X$, then the Dirac measure $\hat{I}{´}_z$ at $z$ is bounded on each compact subset of $C_p(X)$, (2) $z\in \hat{I}ºX$ iff $\hat{I}{´}_z$ is continuous on each compact subset of $C_p(X)$ iff $\hat{I}{´}_z$ is continuous on each compact subset of $C_p^b(X)$, (3) $z\in \upsilon X$ iff $\hat{I}{´}_z$ is bounded on each compact subset of $C_p^b(X)$. It is proved that $\hat{I}ºX$ is the largest subspace $Y$ of ${\hat{I}}^{2}X$ 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.