Closed Extension Space is Irreducible

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 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 $U_1, U_2 \ne \O$.

$\blacksquare$

Proof 2

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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $12$. Closed Extension Topology: $21$