every $\sigma$-compact set is Lindelöf

Theorem 1.

Every $\sigma$-compact (http://planetmath.org/SigmaCompact) set is Lindelöf (every open cover has acountable subcover).

Proof.

Let $X$ be a $\sigma$-compact. Let $𝒜$ be an open cover of$X$ . Since $X$ is $\sigma$-compact, it is the union of countablemany compact sets,

$$X=\bigcup _{i=0}^{\mathrm{\infty }}{X}_i$$

with $X_i$compact. Consider the cover${𝒜}_i=\{A\in 𝒜:X_i\cap A\ne \mathrm{\varnothing }\}$ of the set $X_i$. This cover is well defined, it is not empty and covers $X_i$: for each $x\in X_i$ there is at least one of the open sets $A\in 𝒜$ such that $x\in A$.

Since $X_i$ is compact, the cover${𝒜}_i$ has a finite subcover. Then

$$X_i\subseteq \bigcup _{j=0}^{N_j}{A}_i^j$$

and thus

$$X\subseteq \bigcup _{i=0}^{\mathrm{\infty }}\left(\bigcup _{j=0}^{N_j}{A}_i^j\right).$$

That is, the set$\left\{A_i^j\right\}$ is a countable subcover of $𝒜$that covers $X$.∎