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

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Let (X, d) be a complete metric space. Given a subset A ⊆ X, its Kuratowski measure of non-compactness α(A) ≥ 0 is defined by

 

Note that, if A is itself compact, then α(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 α(A) = 0, then A must be precompact, and indeed compact if A is closed. Also, if A is a subset of B, then α(A) ≤ α(B). In some sense, the quantity α(A) is a numerical description of "how non-compact" the set A is.

Now consider a sequence of sets An ⊆ X, one for each natural number n. Kuratowski's intersection theorem asserts that if these sets are non-empty, closed, decreasingly nested (i.e. An+1 ⊆ An for each n), and α(An) → 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.

References

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  • Kuratowski, Kazimierz (1930). "Sur les espaces complets". Fundamenta Mathematicae. 15: 301–309. doi:10.4064/fm-15-1-301-309.