Trivial Topological Space is Non-Meager

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Theorem

Let $T = \left({S, \tau}\right)$ be a trivial topological space.

Then $T$ is non-meager.


Proof

As $T$ is a trivial topological space, by definition $S$ is a singleton: $S = \left\{{s}\right\}$, say.

Then $\left\{{s}\right\}$ is an open set.

That is, $s$ is an open point.

The result follows from Space with Open Point is Non-Meager.

$\blacksquare$


Sources