Closed Extension Space is Irreducible/Proof 2

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

From Closure of Open Set of Closed Extension Space we have that:

$\forall U \in \tau^*_p: U \ne \O \implies U^- = S$

where $U^-$ is the closure of $U$.

The result then follows by definition of irreducible space.

$\blacksquare$