Let $X$ be a compact (non Haudorff) $T_0$ topological space such that for any subset $\mathcal{A}=\{\mathfrak{x}_\alpha\}_{\alpha\in \Lambda}$ of distinct element of $X$ the set $\{\mathfrak{x}_\beta\}$ is not an open subset of $\mathcal{A}$ for all but finitely many $\beta\in \Lambda$, when considering $\mathcal{A}$ as a subspace of $X$. Is there any description or characterization for such a space?