# Basis for Particular Point Space

## Theorem

Let $T = \left({S, \tau_p}\right)$ 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 $H \in \tau_p$ 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:

$\displaystyle H = \bigcup_{y \mathop \in H} \left\{{y, p}\right\}$

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

$\blacksquare$

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