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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $17$. Either-Or Topology: $3$