## Real Analysis Exchange

- Real Anal. Exchange
- Volume 33, Number 2 (2007), 285-290.

### 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.

#### Article information

**Source**

Real Anal. Exchange, Volume 33, Number 2 (2007), 285-290.

**Dates**

First available in Project Euclid: 18 December 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1229619407

**Mathematical Reviews number (MathSciNet)**

MR2458246

**Zentralblatt MATH identifier**

1161.26001

**Subjects**

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

Secondary: 54D30: Compactness 54D45: Local compactness,$\sigma$-compactness

**Keywords**

continuous function locally compact space lattice

#### Citation

Cater, F. S. Some Lattices of Continuous Functions on Locally Compact Spaces. Real Anal. Exchange 33 (2007), no. 2, 285--290. https://projecteuclid.org/euclid.rae/1229619407