The Urysohn metrization theorem gives conditions which guarantee that a topological space is metrizable. A topological space is metrizable is there is a metric that induces a topology that is equivalent to the topological space itself. These conditions are that the space is regular and second-countable. Regular means that any combination of closed subset and point not in it is separable, and second-countable means there is a countable basis.
The canonical definition of compactness of a topological space 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 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.