In a [[complete metric space]], the following variant of Cantor's intersection theorem holds.
'''Theorem.''' ''Suppose that <math>X</math> is a complete metric space, and <math>(C_k)_{k \geq 1}</math> is a sequence''''of non-empty closed nested subsets of <math>X</math> whose [[diameter]]s tend to zero:''
''Then the intersection of the''''<math>C_k</math> contains exactly one point:''
:<math>\bigcap_{k=1}^\infty C_k = \{x\}</math>
''for some <math>x \in X</math>.''
''Proof (sketch).'' A proof goes as follows. Since the diameters tend to zero, the diameter of the intersection of the ''<math>C_k</math>'' is zero, so it is either empty or consists of a single point. So it is sufficient to show that it is not empty. Pick an element <math>x_k\in C_k</math> for each ''<math>k''</math>. Since the diameter of ''<math>C_k</math>'' tends to zero and the ''<math>C_k</math>'' are nested, the ''<math>x_k</math>'' form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point ''<math>x''</math>. Since each ''<math>C_k</math>'' is closed, and ''<math>x''</math> is a limit of a sequence in ''<math>C_k</math>'', ''<math>x''</math> must lie in ''<math>C_k</math>''. This is true for every ''<math>k''</math>, and therefore the intersection of the ''<math>C_k</math>'' must contain ''<amth>x''</math>. ∎
A converse to this theorem is also true: if ''<math>X''</math> is a metric space with the property that the intersection of any nested family of non-empty closed subsets whose diameters tend to zero is non-empty, then ''<math>X''</math> is a complete metric space. (To prove this, let ''<math>(x_k)_{k \geq 1}</math>'' be a Cauchy sequence in ''<math>X''</math>, and let ''<math>C_k</math>'' be the closure of the tail of this sequence.)
