Arens-Fort Space is not Locally Connected

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Theorem

Let $T = \left({S, \tau}\right)$ be the Arens-Fort space.


Then $T$ is not a locally connected space.


Proof

Let $\mathcal U_0$ be a local basis for $\left({0, 0}\right)$.

Let $U \in \mathcal U_0$.

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

From Clopen Points in Arens-Fort Space, $\left\{ {\left({0, 0}\right)}\right\}$ is not open in $T$.

So $U \ne \left\{ {\left({0, 0}\right)}\right\}$.

Therefore:

$\exists p \in U: p \ne \left({0, 0}\right)$

From Singleton of Element is Subset:

$\left\{ {p}\right\} \subseteq U$

From Clopen Points in Arens-Fort Space it follows that $\left\{ {p}\right\}$ is clopen.

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

$\left({0, 0}\right) \in U$

and so:

$U \ne \left\{ {p}\right\}$

That is:

$\left\{ {p}\right\} \subsetneq U$

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