Uncountable Fort Space is not Separable

Theorem

Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$.

Then $T$ is not a separable space.

Proof

Let $C \subseteq S$ be a countable set.

Since $S$ is uncountable, by Uncountable Set less Countable Set is Uncountable, so is $\relcomp S C$.

Thus there exists some point $x \in \relcomp S C$ and $x \ne p$.

By Clopen Points in Fort Space, $\set x \in \tau_p$.

By Empty Intersection iff Subset of Complement, we have $C \cap \set x = \O$.

Therefore $C$ is not everywhere dense.

Since $C$ is arbitrary, $T$ is not a separable space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $24$. Uncountable Fort Space: $4$