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.
Let (X, d) be a complete metric space. Given a subset A ⊆ X, its Kuratowski measure of non-compactness ñ(A) ≥ 0 is defined by
Note that, if A is itself compact, then ñ(A) = 0, since every cover of A by open balls of arbitrarily small diameter will have a finite subcover. The converse is also true: if ñ(A) = 0, then A must be precompact, and indeed compact if A is closed. Also, if A is a subset of B, then ñ(A) ≤ ñ(B). In some sense, the quantity ñ(A) is a numerical description of "how non-compact" the set A is.
Now consider a sequence of sets A<sub>n</sub> ⊆ X, one for each natural number n. Kuratowski's intersection theorem asserts that if these sets are non-empty, closed, decreasingly nested (i.e. A<sub>n+1</sub> ⊆ A<sub>n</sub> for each n), and ñ(A<sub>n</sub>) â 0 as n â ∞, 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 â ∞, then so must the other two.