mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 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 i ...

. Kuratowski's result is a generalisation of

Cantor's intersection theorem Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets. Topological statement 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 (''X'', ''d'') be a complete metric space. 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

, 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