I was thinking to the following problem. Take a set $X$. If you take a compact topology T (non necessarily Hausdorff) you get the subposet $K_T$ of $\mathcal{P}(X)$ made of compact sets with respect to $T$.

One can list simple properties that $K_T $ respect: it is stable for union, it contains finite sets, it contains X and the empty set. But I don't think these are sufficient.

One possibly useful reformulation is that there exist a "core" $k_T$, which are the closed sets of the topology, with the following property.

$A \in K_T$ iff for every $C_i \in k_T$ with $\bigcap C_i \cap A = \emptyset$, then there exist a finite number of indices $S$ such that $\bigcap_{i \in S} C_i \cap A = \emptyset$.

If you pass to the complement, this is equivalent to the compactness. I was thinking on conditions that ensure that such a "core" exist.