Particular Point Space is Non-Meager/Proof 3

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

$T$ is meager.

By definition, $T$ is meager it is a countable union of subsets of $S$ which are nowhere dense in $T$.

At least one such nowhere dense subset $U$ of $S$ must contain $p$.

By definition, $U$ is nowhere dense in $T$ :
 * $U^-$ contains no open set of $T$ which is non-empty

where $U^-$ denotes the closure of $U$.

By definition of particular point space, $U$ is open in $T$.

By Closure of Open Set of Particular Point Space, $U^- = S$.

But $S$ is itself open in $T$ and non-empty, and so $U$ is not nowhere dense.

From this contradiction it follows that $T$ is non-meager