Existence of Lindelöf Space which is not Sigma-Compact

Theorem

There exists at least one example of a $\sigma$-compact topological space which is not also a compact space.

Proof

Let $T$ be the Sorgenfrey line.

From Sorgenfrey Line is Lindelöf, $T$ is a Lindelöf space.

From Sorgenfrey Line is not $\sigma$-Compact, $T$ is not a $\sigma$-compact space.

Hence the result.

$\blacksquare$

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