Kuratowski's intersection theorem

In mathematics, Kuratowski's intersection theorem is a result in general topology that gives a sufficient condition for a nested sequence of sets to have a non-empty intersection. Kuratowski's result is a generalisation of Cantor's intersection theorem. Whereas Cantor's result requires that the sets involved be compact, Kuratowski's result allows them to be non-compact, but insists that their non-compactness "tends to zero" in an appropriate sense. The theorem is named for the Polish mathematician Kazimierz Kuratowski, who proved it in 1930.

Statement of the theorem

Let be a complete metric space. Given a subset A ⊆ X, its Kuratowski measure of non-compactness α ≥ 0 is defined by
Note that, if A is itself compact, then α = 0, since every cover of A by open balls of arbitrarily small diameter will have a finite subcover. The converse is also true: if α = 0, then A must be precompact, and indeed compact if A is closed. Also, if A is a subset of B, then α ≤ α. In some sense, the quantity α is a numerical description of "how non-compact" the set A is.
Now consider a sequence of sets A_n ⊆ X, one for each natural number n. Kuratowski's intersection theorem asserts that if these sets are non-empty, closed, decreasingly nested, and α → 0 as n → ∞, then their infinite intersection
is a non-empty compact set.
The result also holds if one works with the ball measure of non-compactness or the separation measure of non-compactness, since these three measures of non-compactness are mutually Lipschitz equivalent; if any one of them tends to zero as n → ∞, then so must the other two.