Arens-Fort Space is not Connected
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Theorem
Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a connected space.
Proof
Consider $p \in S$ such that $p \ne \tuple {0, 0}$.
From Clopen Points in Arens-Fort Space, we have that $\set p$ is both open and closed in $T$.
So by definition of closed set, $\relcomp S {\set p}$ is also both open and closed in $T$.
So, by definition, $\set p \mid \relcomp S {\set p}$ is a separation of $T$
Hence the result, by definition of connected space.
$\blacksquare$
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: $7$