Theorem 1   Every $\sigma$ -compact set is Lindelöf (every open cover has a countable subcover).

Proof. Let $X$ be a $\sigma$ -compact. Let $\mathcal{A}$ be an open cover of $X$ . Since $X$ is $\sigma$ -compact, it is the union of countable many compact sets, $$X = \bigcup_{i=0}^{\infty}X_{i}$$ with $X_{i}$ compact. Consider the cover $\mathcal{A}_{i}=\left\{A\in\mathcal{A}: X_{i}\cap A \neq \emptyset \right\}$ 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 $A\in\mathcal{A}$ such that $x \in A$ .

Since $X_{i}$ is compact, the cover $\mathcal{A}_{i}$ has a finite subcover. Then $$X_{i}\subseteq \bigcup_{j=0}^{N_j}A_{i}^{j}$$ and thus $$X\subseteq \bigcup_{i=0}^{\infty} \left(\bigcup_{j=0}^{N_j}A_{i}^{j}\right).$$ That is, the set $\left\{A_{i}^{j} \right\}$ is a countable subcover of $\mathcal{A}$ that covers $X$ .