Sets in Modified Fort Space are Disconnected

Theorem
Let $T = \struct {S, \tau_{a, b}}$ be a modified Fort space.

Let $H$ be a subset of $S$ with more than one point.

Then $H$ is disconnected.

Proof
By Isolated Points in Subsets of Modified Fort Space:


 * $\exists x \in H: x$ is isolated

By Point in Topological Space is Open iff Isolated, $\set x$ is open in $T$.

By Modified Fort Space is $T_1$ and definition of $T_1$ space, $\set x$ is closed in $T$.

Therefore $\relcomp S {\set x}$ is open in $T$.

Then we have:

This shows that $H$ is disconnected.