Basis for Particular Point Space
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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 $T$.
$\blacksquare$
It could equally well be shown that $\BB$ is also a synthetic basis for $T$.