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 a right half-open interval space.

From Right Half-Open Interval Space is Lindelöf, $T$ is a Lindelöf space.

From Right Half-Open Interval Space is not $\sigma$-Compact, $T$ is not a $\sigma$-compact space.

Hence the result.