Basis for Particular Point Space

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

Consider the set $\mathcal B$ defined as:
 * $\mathcal B = \left\{{\left\{{x, p}\right\}: x \in S}\right\} \cup \left\{{p}\right\}$

Then $B$ is a basis for $S$.

Proof
Let $H \in \vartheta_2$ be open in $T$.

Then:
 * $\forall y \in H: \exists \left\{{y, p}\right\} \in \mathcal B$

which also holds when $y = p$ as $\left\{{y, p}\right\} = \left\{{p}\right\} \in \mathcal B$.

Thus:
 * $\displaystyle H = \bigcup_{y \in H} \left\{{y, p}\right\}$

So $\mathcal B$ is an analytic basis for $T$.

It could equally well be shown that $\mathcal B$ is also a synthetic basis for $T$.