Basis for Particular Point Space

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

Consider the set $\BB$ defined as:
 * $\BB = \set {\set {x, p}: x \in S} \cup \set p$

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

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

Then:
 * $\forall y \in H: \exists \set {y, p} \in \BB$

which also holds when $y = p$ as $\set {y, p} = \set p \in \BB$.

Thus:
 * $\ds H = \bigcup_{y \mathop \in H} \set {y, p}$

So $\BB$ is an analytic basis for $T$.

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