# What properties do compact sets satisfy?

Let $$X$$ be a set, and $$\mathcal{C}$$ be a set of subsets of $$X$$ which contains the empty set and is closed under finite unions and infinite intersections.

Is it true that there must exist a topology on $$X$$ such that the compact sets of $$X$$ are $$\mathcal{C}$$? If not, what other properties does $$\mathcal{C}$$ need to satisfy for this to be true?