Trivial Topological Space is Non-Meager

Theorem

Let $T = \struct {S, \tau}$ be a trivial topological space.

Then $T$ is non-meager.

Proof

As $T$ is a trivial topological space, by definition $S$ is a singleton: $S = \set s$, say.

Then $\set s$ is an open set.

That is, $s$ is an open point.

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

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Problems: Section $1: \ 4$