Arens-Fort Space is Non-Meager

Theorem

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

Then $T$ is a non-meager space.

Proof

From Meager Sets in Arens-Fort Space, we have that $A \subseteq S$ is meager in $T$ if and only if $A = \set {\tuple {0, 0} }$.

So as $\set {\tuple{0, 0} } \ne S \subseteq S$, it follows that $T$ is non-meager.

$\blacksquare$

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: $8$