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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $7$. Deleted Integer Topology: $5$