Odd-Even Topology is Second-Countable

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 second-countable.

Hence $T$ is also first-countable, separable and Lindelöf.

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

is a basis for $T$.

There is an obvious one-to-one correspondence $\phi: \Z_{>0} \leftrightarrow \mathcal B$ between $\Z_{>0}$ and $\mathcal B$:
 * $\forall x \in \Z_{>0}: \phi \left({x}\right) = \left\{{2 x - 1, 2 x}\right\}$

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

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

Thus $\mathcal B$ is also countable by definition of countability.

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

Then we have that a Second-Countable Space is First-Countable, a Second-Countable Space is Separable and a Second-Countable Space is Lindelöf.