Arens-Fort Space is not Connected

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Theorem

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


Then $T$ is not a connected space.


Proof

Consider $p \in S$ such that $p \ne \left({0, 0}\right)$.

From Clopen Points in Arens-Fort Space, we have that $\left\{{p}\right\}$ is both open and closed in $T$.

So by definition of closed set, $\complement_S \left({\left\{{p}\right\}}\right)$ is also both open and closed in $T$.

So, by definition, $\left\{{p}\right\} \mid \complement_S \left({\left\{{p}\right\}}\right)$ is a separation of $T$

Hence the result, by definition of connected space.

$\blacksquare$


Sources