Particular Point Space is Non-Meager

Theorem
Let $T = \left({S, \tau_p}\right)$ be a particular point space.

Then $T$ is of the second category.

Proof 1
Suppose $T$ were first category.

Then it would be the union of a countable set 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 null.

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

Hence $T$ is not first category and so by definition must be of the second category.

Proof 2
By definition of particular point space, any subset of $S$ which contains $p$ is open in $T$.

So $\left\{{p}\right\}$ itself is open in $T$.

That is, $p$ is an open point.

The result follows from Space with Open Point is Second Category.