Arens-Fort Space is not Extremally Disconnected

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Theorem

Let $T = \struct {S, \tau}$ be the Arens-Fort space.


Then $T$ is not extremally disconnected.


Proof

Let $\map {S_m} V$ denote the set:

$\map {S_m} V := \set {n: \tuple {m, n} \notin V}$

where $V \subseteq \Z_{\ge 0} \times \Z_{\ge 0}$ (the same set $S_m$ used in the definition of the Arens-Fort space).

Let $U = \set {\tuple {n, m}: \exists k: m = 2 k} \setminus \set {\tuple {0, 0} }$.

From the definition of the Arens-Fort space, $U$ is open in $T$ since $\left({0, 0}\right) \notin U$.

We have that:

$\tuple {0, 0} \in \relcomp S U$

and:

$\map {S_m} U$ is infinite for any $m \in \N$.

Thus by definition of the Arens-Fort space, $\relcomp S U$ is not open in $T$.

So $U$ is not closed in $T$.


Let $U^-$ denote the closure of $U$.

We have that:

$\tuple {0, 0} \notin \relcomp S {U \cup \set {\tuple {0, 0} } }$

Therefore by definition of the Arens-Fort space, $\relcomp S {U \cup \set {\tuple {0, 0} } }$ is open in $T$.

Therefore $U \cup \set {\tuple {0, 0} }$ is closed in $T$.

From Set Closure is Smallest Closed Set, it follows that $U^- = U \cup \set {\tuple {0, 0} }$.


We have that:

$\tuple {0, 0} \in U \cup \set {\tuple {0, 0} }$

and:

$\map {S_m} {U \cup \set {\tuple {0, 0} } }$ is infinite for any $m \in \N$.

Therefore by definition of the Arens-Fort space, $U \cup \set {\tuple {0, 0} }$ is not open.


Thus we have created an open set $U$ whose closure $U^-$ is not itself open.

Thus by definition the Arens-Fort space is not extremally disconnected.

$\blacksquare$

Sources