# 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

$\exists x \in H: x$ 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:

 $\ds H$ $\subseteq$ $\ds S$ $\ds$ $=$ $\ds \set x \cup \relcomp S {\set x}$ Union with Relative Complement $\ds H \cap \set x \cap \relcomp S {\set x}$ $\subseteq$ $\ds \set x \cap \relcomp S {\set x}$ Intersection is Subset $\ds$ $=$ $\ds \O$ Intersection with Relative Complement is Empty $\ds \leadsto \ \$ $\ds H \cap \set x \cap \relcomp S {\set x}$ $=$ $\ds \O$ Subset of Empty Set $\ds H \cap \set x$ $=$ $\ds \set x$ $\ds$ $\ne$ $\ds \O$ $\ds H \cap \relcomp S {\set x}$ $=$ $\ds H \setminus \set x$ Set Difference as Intersection with Relative Complement $\ds$ $\ne$ $\ds \O$ $H$ has more than one point

This shows that $H$ is disconnected.

$\blacksquare$