Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the closure of $\{x_i\}_{i \not=j}$.

Is there any characterization of such spaces? Or is there any sufficient or necessary condition for this property? Thanks for any help.