Existence of Sigma-Compact Space which is not 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 real number line under the Euclidean topology.

From Real Number Space is Sigma-Compact, $T$ is a $\sigma$-compact space.

From Real Number Space is not Countably Compact, $T$ is not a countably compact space.

The result follows from Compact Space is Countably Compact.