Space with Open Point is Non-Meager

Theorem

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

Let $x \in S$ be an open point.

Then $T$ is a non-meager space.

Proof

Let $x \in S$ be an open point of $T$.

That is:

$\set x \in \tau$

Recall that:

a topological space is non-meager if it is not meager

and:

a topological space is meager if and only if it is a countable union of subsets of $S$ which are nowhere dense in $S$.

Aiming for a contradiction, suppose that $T$ is meager.

Let:

$\ds T = \bigcup \SS$

where $\SS$ is a countable set of subsets of $S$ which are nowhere dense in $S$.

Then:

$\exists H \in \S S: x \in H$

and so:

$\set x \subseteq H$

We have that $H$ is nowhere dense in $T$.

By definition, its closure $H^-$ contains no open set of $T$ which is non-empty.

But from Set is Subset of its Topological Closure we have that:

$H \subseteq H^-$

So by Subset Relation is Transitive:

$\set x \subseteq H^-$

So $H$ is not nowhere dense.

Therefore $T$ cannot be a countable union of subsets of $S$ which are nowhere dense in $S$.

That is, $T$ is not meager.

Hence the result by definition of non-meager.

$\blacksquare$

Sources

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