# Another characterization of compactness

Originally published at 狗和留美者不得入内. You can comment here or there.

The canonical definition of compactness of a topological space $X$ is every open cover has finite sub-cover. We can via contraposition translate this to every family of open sets with no finite subfamily that covers $X$ is not a cover. Not a cover via de Morgan’s laws can be characterized equivalently as has complements (which are all closed sets) which have finite intersection. The product is:

A topological space is compact iff for every family of closed sets with the finite intersection property, the intersection of that family is non-empty.

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