Either-Or Topology is not Separable

Theorem
Let $T = \struct {S, \tau}$ be the either-or space.

Then $T$ is not a separable space.

Proof
From Limit Points of Either-Or Topology, the only limit point of any set of $S$ is $0$.

So the only set whose closure is $S$ are $S \setminus \set 0$ and $S$ itself.

So these two are the only subsets of $S$ which are everywhere dense in $S$.

Both of these are uncountable.

Hence the result, by definition of separable space.