I have recently encountered this result.
- Let $X$ be $\sigma$-compact, locally compact Hausdorff space and $\mu$ is a Radon measure on $X$. Then the space of continuous functions with compact support is dense in that of $\mu$-integrable functions w.r.t. $\|\cdot\|_{L_1}$. ref
I got that
- Polish space is not necessarily $\sigma$-compact. ref
- $\sigma$-compact Polish space is not necessarily locally compact. ref
I would like to ask if the following result holds, i.e.,
Locally compact Polish space is $\sigma$-compact.
Update:
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As proven in an below answer: locally-compact second-countable space is $\sigma$-compact.
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From this, every locally-compact second-countable Hausdorff space is a Polish space.
Best Answer
Yes, a locally compact Polish space is $\sigma$-compact.
Since it is Polish, such a space admits a countable base $\mathcal{U} = \{B_i\}_{i=1}^n$ of open sets for the entire space; by local compactness, each element in $\mathcal{U}$ can be taken so that $\overline{B_i}$ is compact. Since $X = \bigcup_{n=1}^\infty \overline{B_i}$, we have exhibited the $\sigma$-compactness of $X$.