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 if and only if 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.

$\blacksquare$

Also see

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties