Some Lattices of Continuous Functions on Locally Compact Spaces

Abstract

Let $U$ be a locally compact Hausdorff space that is not compact. Let $L(U)$ denote the family of continuous real valued functions on $U$ such that for each $f\in L(U)$ there is a nonzero number $p$ (depending on $f$) for which $f\!-\!p$ vanishes at infinity. Then $L(U)$ is obviously a lattice under the usual ordering of functions. \par In this paper we prove that $L(U)$, as a lattice alone, characterizes the locally compact space $U$. \par Let $S$ be a locally compact Hausdorff space. Define $T(S)$ to be $L(S)$ if $S$ is not compact, and $T(S)$ to be $C(S)$ if $S$ is compact. We prove that any locally compact Hausdorff spaces $S_1$ and $S_2$ are homeomorphic if and only if their associated lattices $T(S_1)$ and $T(S_2)$ are isomorphic.