`Let $p$ be the non-isolated point of an almost discrete space.
Any neighborhood of $p$ is closed.
Any point $x\neq p$ has a smallest neighborhood $\{x\}$, which must exist in any basis.
The closure of such a neighborhood must be either $\{x\}$ or $\{x,p\}$.
Therefore, if the space is not semiregular, there must exist some $y$ such that
$\operatorname{cl}\{y\}=\{y,p\}$ and $\operatorname{in}\{y,p\}=\{y,p\}$,
giving us a finite neighborhood of $p$.
Note that an {P203} space that is not {P10} must be
the disjoint union of the {S10} and a {P52} space.`.length