Right Order Topology on Strictly Positive Integers is not Metrizable

Theorem
Let $\Z_{>0}$ be the set of strictly positive integers.

Let $T = \struct {\Z_{>0}, \tau}$ denote the right order space on $\Z_{>0}$.

Then $T = \struct {\Z_{>0}, \tau}$ is not a metrizable space.

Proof
Let $m, n \in \Z_{>0}$ such that $m < n$.

Let $O_m$ and $O_n$ be arbitrary non-empty open sets of $T$.

Then:
 * $O_m \cap O_n = O_m$

As $O_m$ and $O_n$ are arbitrary, it follows that there exist no $O_m$ and $O_n$ in $\tau$ such that $O_m \cap O_n = \O$.

Hence $T$ is not Hausdorff.

The result follows from Metrizable Space is Hausdorff.