Set of Integers is not Compact

Theorem

Let $\Z$ be the set of integers.

Then $\Z$ is not compact.

Proof

Let $\R$ be the real number line considered as an Euclidean space.

From Set of Integers is not Bounded, $\Z$ is not bounded in $\R$.

The result follows by definition of compact.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness