Deleted Integer Topology is Second-Countable

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the deleted integer topology on the set $S = \R_+ \setminus \Z$.

Then $T$ is second-countable.

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

Proof
Let $\Z^*_+$ be understood as the set of strictly positive integers:
 * $\Z^*_+ = \left\{{x \in \Z: x > 0}\right\} = \left\{{1, 2, 3, \ldots}\right\}$

From Basis for Partition Topology, the set:
 * $\mathcal B = \left\{{\left({n - 1 . . n}\right): n \in \Z^*_+}\right\}$

is a basis for $T$.

There is an obvious one-to-one correspondence $\phi: \Z^*_+ \leftrightarrow \mathcal B$ between $\Z^*_+$ and $\mathcal B$:
 * $\forall x \in \Z^*_+: \phi \left({x}\right) = \left({x - 1 . . x}\right)$

But $\Z^*_+ \subseteq \Z$ and Integers are Countable.

So from Subset of Countable Set, $\Z^*_+$ 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 Second-Countable Space is First-Countable, Second-Countable Space is Separable and Second-Countable Space is Lindelöf.