Sigma-Compact Space is Lindelöf

Theorem
Every $\sigma$-compact space is a Lindelöf space.

Proof
Let $T = \left({S, \tau}\right)$ be a $\sigma$-compact space.

By definition:
 * $T$ is a Lindelöf space every open cover of $X$ has a countable subcover.

By definition of $\sigma$-compact space, $T = \bigcap \mathcal T$ where $\mathcal T$ is the union of countably many compact subspaces.

Let $\mathcal C$ be an open cover of $T$.

Each element of $\mathcal T$ is covered by a finite number of elements of $\mathcal C$.

Hence $T$ is covered by a countable union of a finite number of elements of $\mathcal C$.

Hence $\mathcal C$ has a countable subcover.

Hence the result.

Also see

 * Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf