Deleted Integer Topology is Separable

Corollary to Deleted Integer Topology is Second-Countable
Let $S = \R_{\ge 0} \setminus \Z$.

Let $\tau$ be the deleted integer topology on $S$.

The topological space $T = \struct {S, \tau}$ is separable.

Proof
From Deleted Integer Topology is Second-Countable, $T$ is second-countable.

The result follows from Second-Countable Space is Separable.