Odd-Even Topology is not Countably Compact

Theorem
Let $T = \left({\Z_{>0}, \tau}\right)$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.

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 $T$:
 * $\mathcal P = \left\{{\left\{{2 k - 1, 2 k}\right\}: k \in \Z_{>0} }\right\}$

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

Hence the result.