Particular Point Topology is Topology

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be a particular point space.

Then $\vartheta_p$ is a topology on $S$, and $T$ is a topological space.

Proof
We have by definition that $\varnothing \in \vartheta_p$, and as $p \in S$ we have that $S \in \vartheta_p$.

Now let $U_1, U_2 \in \vartheta_p$.

By definition $p \in U_1$ and $p \in U_2$, and so $p \in U_1 \cap U_2$ by definition of set intersection.

So $U_1 \cap U_2 \in \vartheta_p$.

Now let $\mathcal U \subseteq \vartheta_p$.

We have that $\forall U \in \mathcal U: p \in U$.

Hence from Subset of Union $p \in \bigcup \mathcal U$.

So all the properties are fulfilled for $\vartheta_p$ to be a topology on $S$.