Odd-Even Topology is Second-Countable

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the odd-even topology on the strictly positive integers $S$.

Then $T$ is second-countable.

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

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

is a basis for $T$.

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

But $S \subseteq \Z$ and Integers are Countable.

So from Subset of Countable Set, $S$ is countable.

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

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

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