Countably compact $T_0$ space $S$ such that for all $X\subseteq S$ and every $p\in X'$, there exists an open set $O$ such that ...

Question

Is there an infinite $T_0$ space $S$ such that every infinite subset has a limit point, and such that, for all $X\subseteq S$ and every limit point $p$ of $X$, there exists an open set $O$ such that $p\in O$ and $O\cap X$ is finite?

Answer 1

Note that such a space is locally finite: every point is either not a limit point of the whole space and is thus isolated, or is a limit point and by assumption has a finite neighborhood.

Here's one such example: the left-ray topology on $\{0,1,2,\dots\}$, with open sets of the form $\{0,1,\dots,n-1\}$. Then every nonempty set has a limit point: let $m$ belong to the set, then $m+1$ is a limit. Furthermore, every point has a finite neighborhood, so we're done. (Per the comment I made in the question, this space is $T_0$ but not $T_1$.)