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 $T = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

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

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

The result follows from Compact Space is Countably Compact.

$\blacksquare$

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