Continuous compactly supported real valued functions on a locally compact and $\sigma$ compact space is separable

Asked Apr 17 '14 at 20:28

Active Apr 17 '14 at 23:44

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I already know that if $X$ is a compact metric space then the space of continuous real valued functions $C(X \to \mathbb R)$ are separable. What I'm trying to prove that if $X$ is a locally compact metric space and is $\sigma$-compact. The space of continuous compactly supported functions $C_c( X \to \mathbb R)$ are separable.

Here's my approach: As $X$ is $\sigma$-compact, we can write $X = \bigcup_{ n=1}^\infty K_n$, where $K_n$ are compact and increasing. Then since each $C(K_n \to \mathbb R)$ is separable, it suffices to show that the set $ \{ f \in C_c (X \to \mathbb R) : \operatorname{supp}(f) \subset K_n \}$ is dense in $C(X \to \mathbb R)$ (Not sure). Then get I stuck.

Any help is appreciated.

edited Apr 17 '14 at 23:44

kahen

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asked Apr 17 '14 at 20:28

user112564

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If $X$ is compact, then $C(X\to \mathbb{R})$ need not be separable. Is $X$ perhaps required to be a metric space? – Daniel Fischer Apr 17 '14 at 20:45
@DanielFischer Oh, yes. You are right. Fixed. – user112564 Apr 17 '14 at 23:37
I think "$\sigma$-locally compact" is a common abbreviation for "locally compact and $\sigma$-compact". I find it rolls off the tongue more easily. – kahen Apr 17 '14 at 23:43
Next question: which topology do you take on $C_c(X)$? Uniform convergence? – Daniel Fischer Apr 17 '14 at 23:44
@DanielFischer Yes. The supnorm. – user112564 Apr 17 '14 at 23:46
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Are you familiar with Alexandrov compactifications (one-point-compactifications)? – Daniel Fischer Apr 17 '14 at 23:58
@DanielFischer Never heard about this. Sorry... – user112564 Apr 17 '14 at 23:59
Pity. That would have made it easy. – Daniel Fischer Apr 18 '14 at 00:00

Continuous compactly supported real valued functions on a locally compact and $\sigma$ compact space is separable

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