Arens-Fort Space is not Locally Connected

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Theorem

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


Then $T$ is not a locally connected space.


Proof

Let $\mathcal U_0$ be a local basis for $\tuple {0, 0}$.

Let $U \in \mathcal U_0$.

By definition of local basis, $U$ is open in $T$.

From Clopen Points in Arens-Fort Space, $\set {\tuple {0, 0} }$ is not open in $T$.

So $U \ne \set {\tuple {0, 0} }$.

Therefore:

$\exists p \in U: p \ne \tuple {0, 0}$

From Singleton of Element is Subset:

$\set p \subseteq U$

From Clopen Points in Arens-Fort Space it follows that $\set p$ is clopen.

As $U \in U_0$ it follows by definition of local basis that:

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

and so:

$U \ne \set p$

That is:

$\set p \subsetneq U$

From Connected iff no Proper Clopen Sets, the set $U$ is not connected.



It is deduced that any local basis is formed with disconnected sets.

Thus, by definition, $T$ is not locally connected.

$\blacksquare$


Sources