Neighborhood of Origin of Arens-Fort Space is Closed

Theorem

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

Every neighborhood of $\tuple {0, 0}$ is closed in $T$.

Proof

Let $H \subseteq S$ such that:

$\exists U \in \tau: \tuple {0, 0} \in U \subseteq H \subseteq S$

that is: such that $H$ is a neighborhood of $\tuple {0, 0}$ in $T$.

As $\tuple {0, 0} \in H$ it follows that $\tuple {0, 0} \notin \relcomp S H$.

So, by definition of the Arens-Fort space, $\relcomp S H$ is open in $T$.

So by definition, we have that $H$ is closed.

$\blacksquare$

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: $8$