Closed Extension Space is Irreducible

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

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


Then $T^*_p$ is irreducible.


Proof 1

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 $U_1, U_2 \ne \varnothing$.

$\blacksquare$


Proof 2

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

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

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

The result then follows by definition of irreducible space.

$\blacksquare$


Sources