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2007/2008 Some Lattices of Continuous Functions on Locally Compact Spaces
F. S. Cater
Real Anal. Exchange 33(2): 285-290 (2007/2008).


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.


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F. S. Cater. "Some Lattices of Continuous Functions on Locally Compact Spaces." Real Anal. Exchange 33 (2) 285 - 290, 2007/2008.


Published: 2007/2008
First available in Project Euclid: 18 December 2008

zbMATH: 1161.26001
MathSciNet: MR2458246

Primary: 26A15
Secondary: 54D30 , 54D45

Keywords: Continuous function , lattice , locally compact space

Rights: Copyright © 2007 Michigan State University Press


Vol.33 • No. 2 • 2007/2008
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