Odd-Even Topology is Weakly 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 weakly countably compact.

Proof
Let $H \subseteq \Z_{>0}$ such that $H$ is infinite.

Let $x \in H$.

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

So $U$ is a union of sets of the form $\set {2 k - 1, 2 k}$.

Now if $x \in U$, it will be of the form $2 k - 1$ or $2 k$.

So there will exist $y \in U$ of the form $2 k$ or $2 k - 1$.

So, by definition, $x$ is a limit point of $H$.

So, by definition, $T$ is weakly countably compact.