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?


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