When is $C_0(X)$ separable?

Solution 1:

Your argument is correct, thanks to the following result.

Theorem. For a locally compact Hausdorff space $X$ the following are equivalent:

1. $X$ is Lindelöf.
2. $X$ is $\sigma$-compact.
3. There is a sequence $\langle K_n:n\in\Bbb N\rangle$ of compact subspaces of $X$ such that $K_n\subseteq\operatorname{int}K_{n+1}$ for each $n\in\Bbb N$ and $X=\bigcup_{n\in\Bbb N}K_n$.
4. The point at infinity in the one-point compactification of $X$ has a countable local base.

Proof. Suppose that $X$ is Lindelöf. For each $x\in X$ let $U_x$ be an open nbhd of $x$ with compact closure, and let $\mathscr{U}=\{U_x:x\in X\}$. $\mathscr{U}$ is an open cover of $X$, so it has a countable subcover $\{V_n:n\in\Bbb N\}$. Clearly $X=\bigcup_{n\in\Bbb N}\operatorname{cl}V_n$ is a representation of $X$ as a countable union of compact sets, so $X$ is $\sigma$-compact.

Now assume that $X$ is $\sigma$-compact, and let $\{C_n:n\in\Bbb N\}$ be a countable cover of $X$ by compact sets. Let $K_0=C_0$. If the compact set $K_n$ has already been defined, for each $x\in K_n$ let $U_x$ be an open nbhd of $x$ with compact closure, and let $\mathscr{U}=\{U_x:x\in K_n\}$. $K_n$ is compact, so $\mathscr{U}$ has a finite subcover $\{V_1,\dots,V_m\}$. Let $K_{n+1}=C_n\cup\bigcup_{k=1}^m\operatorname{cl}V_k$; then $K_{n+1}$ is compact, and $K_n\subseteq\operatorname{int}K_{n+1}$. Finally, $C_n\subseteq K_n$ for $n\in\Bbb N$, so $\bigcup_{n\in\Bbb N}K_n=X$.

Now assume that there is a sequence $\langle K_n:n\in\Bbb N\rangle$ as in (3). Let $X^*$ be the one-point compactification of $X$, and let $p$ be the point at infinity. For $n\in\Bbb N$ let $B_n=X^*\setminus K_n$, and let $\mathscr{B}=\{B_n:n\in\Bbb N\}$; then $\mathscr{B}$ is a countable local base at $p$. To see this, let $U$ be any open nbhd of $p$ in $X^*$. Then $C=X^*\setminus U$ is a compact subset of $X$. $\bigcup_{n\in\Bbb N}\operatorname{int}K_n=X$, so $\{\operatorname{int}K_n:n\in\Bbb N\}$ is an increasing open cover of $C$, and therefore there is an $n\in\Bbb N$ such that $C\subseteq\operatorname{int}K_n\subseteq K_n$, whence $B_n\subseteq U$.

Finally, suppose that $X^*$ is first countable at $p$, the point at infinity, and let $\mathscr{B}=\{B_n:n\in\Bbb N\}$ be a nested local base at $p$. For $n\in\Bbb N$ let $K_n=X^*\setminus B_n$; then each $K_n$ is compact, and $X=\bigcup_{n\in\Bbb N}K_n$. Let $\mathscr{U}$ be an open cover of $X$. For each $n\in\Bbb N$ let $\mathscr{U}_n$ be a finite subset of $\mathscr{U}$ covering the compact set $K_n$, and let $\mathscr{V}=\bigcup_{n\in\Bbb N}\mathscr{U}_n$; then $\mathscr{V}$ is a countable subcover of $\mathscr{U}$. $\dashv$

Of course a second countable space is Lindelöf, so you get all of (1)-(4).