Arens-Fort Space is not Locally Connected

Theorem

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

Then $T$ is not a locally connected space.

Proof

Let $\UU_0$ be a local basis for $\tuple {0, 0}$.

Let $U \in \UU_0$.

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.

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

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

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $7$