Particular Point Space is Non-Meager/Proof 1

Theorem

Let $T = \struct {S, \tau_p}$ be a particular point space.

Then $T$ is non-meager.

Proof

Suppose $T$ were meager.

Then it would be a countable union of subsets which are nowhere dense in $T$.

Let $H \subseteq S$.

From Closure of Open Set of Particular Point Space, the closure of $H$ is $S$.

From the definition of interior, the interior of $S$ is $S$.

So the interior of the closure of $H$ is not empty.

So $T$ can not be the union of a countable set of subsets which are nowhere dense in $T$.

Hence $T$ is not meager and so by definition must be non-meager.

$\blacksquare$