Fort Space is T1

Theorem

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

Then $T$ is a $T_1$ (Fréchet) space.

Proof

From Fort Space is Excluded Point Space with Finite Complement Space, $T$ is an expansion of a finite complement space.

Then we have that a Finite Complement Space is $T_1$.

Then from Separation Properties Preserved by Expansion we have that as a finite complement space is a $T_1$ space, then so is $T$.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $23 \text { - } 24$. Fort Space: $1$