Subset of Particular Point Space is either Open or Closed

Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.

Let $H \subseteq S$ be any subset of $T$.

Then $H$ is either open or closed in $T$.

The only sets which are both closed and open in $T$ are $S$ and $\O$.

Proof
Let $H \subseteq S$.

There are two cases to consider:
 * $p \in H$
 * $p \notin H$

If $p \in H$ then by definition of a particular point topology, $H$ is open.

If $p \notin H$, then $p \in \relcomp S H$, where $\relcomp S H$ is the relative complement of $H$ in $S$.

So $\relcomp S H$ is open by definition of a particular point topology.

From the definition of a closed set, $\relcomp S {\relcomp S H}$ is closed in $T$.

From Relative Complement of Relative Complement we have that $\relcomp S {\relcomp S H} = H$ and so $H$ is closed in $T$.

Now suppose $H \subseteq T$ is both closed and open in $T$.

From Open and Closed Sets in Topological Space, if $H = \O$ or $H = S$ then H is both closed and open in $T$.

So, suppose $H \ne \O$ or $H \ne S$.

As $H \subseteq T$ is both closed and open in $T$, $\relcomp S H$ is also both closed and open in $T$.

From Boundary is Intersection of Closure with Closure of Complement we have:
 * $\partial H = H^- \cap \paren {\relcomp S H}^-$

where $H^-$ is the closure of $H$.

By Closure of Open Set of Particular Point Space we have that $H^- = S$, and of course $\paren {\relcomp S H}^- = S$.

So:


 * $\partial H = H^- \cap \paren {\relcomp S H}^- = S$

However, from Set Clopen iff Boundary is Empty we have that $\partial H = \O$.

Hence if $H$ is both closed and open in $T$, then $H = \O$ or $H = S$.