Odd-Even Topology is not Countably Compact

Theorem
Let $T = \struct {\Z_{>0}, \tau}$ 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 $\PP$ be the partition which is the basis for $T$:
 * $\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$

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

Hence the result.