Sigma-Compact Space is Lindelöf

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

Proof
Let $T = \struct {S, \tau}$ 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, $T = \bigcap \TT$ where $\TT$ is the union of countably many compact subspaces.

Let $\CC$ be an open cover of $T$.

Each element of $\TT$ is covered by a finite number of elements of $\CC$.

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

Hence $\CC$ has a countable subcover.

Hence the result.

Also see

 * Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf