Let me formally ask the question again.
Question. Is there a $T_{0}$-space that has a subset $A \subseteq X$ that has the folloing property?
There exists $p \in X$, a limit point of $A$ such that there is a neighborhood of $p$ intersects $A$ at finitely many points.
Note that there is no such subset $A$ in $T_{1}$ space.