Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

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.

Tags: geometry/topology, point-set topology
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