Closed Extension Space is Irreducible/Proof 1

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.

Then $T^*_p$ is irreducible.

Proof

Trivially, by definition, every open set in $T^*_p$ contains $p$.

So:

$\forall U_1, U_2 \in \tau^*_p: p \in U_1 \cap U_2$

for $U_1, U_2 \ne \O$.

$\blacksquare$