Odd-Even Topology is not Countably Compact

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the odd-even topology on the strictly positive integers $S$.

Then $T$ is not countably compact.

Proof
By definition, the odd-even topology is a partition topology.

Let $\mathcal P$ be the partition which is the basis for $T$:
 * $\mathcal P = \left\{{\left\{{2k - 1, 2k}\right\}: k \in S}\right\}$

Then $\mathcal P$ is a countable open cover of $S$ which has no finite subcover.

Hence the result.