Subset of Particular Point Space is either Open or Closed

Theorem
Let $T = \left({S, \vartheta_p}\right)$ 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 $\varnothing$.

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 $p \in \complement_S \left({H}\right)$, where $\complement_S \left({H}\right)$ is the relative complement of $H$ in $S$.

So $\complement_S \left({H}\right)$ is open by definition of a particular point topology.

But from the definition of the closed set, $\complement_S \left({\complement_S \left({H}\right)}\right)$ is closed in $T$.

But from Relative Complement of Relative Complement we have that $\complement_S \left({\complement_S \left({H}\right)}\right) = H$ and so $H$ is closed in $T$.

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

We know from Open and Closed Sets in a Topological Space that if $H = \varnothing$ or $H = S$ then H is both closed and open in $T$.

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

As $H \subseteq T$ is both closed and open in $T$, $\complement_S \left({H}\right)$ is also both closed and open in $T$.

From Boundary is Intersection of Closure with Closure of Complement we have:
 * $\partial H = H^- \cap \left({\complement_S \left({H}\right)}\right)^-$

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

By Closure of Open Set of Particular Point Space we have that $H^- = S$, and of course $\left({\complement_S \left({H}\right)}\right)^- = S$.

So:


 * $\partial H = H^- \cap \left({\complement_S \left({H}\right)}\right)^- = S$

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

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