# 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.

$\blacksquare$