mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Kuratowski's intersection theorem is a result in

general topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...

that gives a sufficient condition for a nested sequence of sets to have a non-empty

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

. Kuratowski's result is a generalisation of Cantor's intersection theorem. Whereas Cantor's result requires that the sets involved be

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

, 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 Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Polish people, people from Poland or of Polish descent * Polish chicken * Polish brothers (Mark Polish and Michael Polish, born 1970), American twin ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Ma ...

, who proved it in 1930.

Statement of the theorem

Let (''X'', ''d'') be a

complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...

. Given a subset ''A'' ⊆ ''X'', its Kuratowski measure of non-compactness ''α''(''A'') ≥ 0 is defined by :

\alpha(A) = \inf \left\.

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 ''A''_''n'' ⊆ ''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''_''n''+1 ⊆ ''A''_''n'' for each ''n''), and ''α''(''A''_''n'') → 0 as ''n'' → ∞, then their infinite intersection :

\bigcap_ A_

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

* {{cite journal , last = Kuratowski , first = Kazimierz , authorlink = Kazimierz Kuratowski , title = Sur les espaces complets , journal = Fundamenta Mathematicae , volume = 15 , year = 1930 , pages = 301–309 , doi = 10.4064/fm-15-1-301-309 , doi-access = free Compactness theorems