 Cantor's intersection theorem

A metric space is complete if and only if every decreasing sequence of nonempty closed sets whose diameters tend to zero has nonempty intersection. In that case the intersection contains exactly one point.

Cantor's intersection theorem is a result in set theory that pertains to nested sequences of closed sets in a complete metric space. The theorem states that if you have a sequence of closed sets in a complete metric space such that each set is contained within the previous one (i.e., a nested sequence), and if the size of these sets shrinks down to a single point, then the intersection of all these sets is non-empty and contains exactly one point.