Odd-Even Topology is Second-Countable

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 second-countable.

Proof
From Basis for Partition Topology, the set:
 * $\BB := \set {\set {2 k - 1, 2 k}: k \in \Z, k > 0}$

is a basis for $T$.

There is an obvious one-to-one correspondence $\phi: \Z_{>0} \leftrightarrow \BB$ between $\Z_{>0}$ and $\BB$:
 * $\forall x \in \Z_{>0}: \map \phi x = \set {2 x - 1, 2 x}$

But $\Z_{>0} \subseteq \Z$, and Integers are Countably Infinite.

So from Subset of Countably Infinite Set is Countable, $\Z_{>0}$ is countable.

Thus $\BB$ is also countable by definition of countability.

So we have that $T$ has a countable basis, and so is second-countable by definition.