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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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    Note that compact sets need not be closed under all intersections in general. Closed sets are. Even with two compact sets this can fail. – Henno Brandsma Feb 25 '20 at 12:11
  • https://math.stackexchange.com/questions/229792/intersection-of-finite-number-of-compact-sets-is-compact – Moishe Kohan Feb 26 '20 at 02:28

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