Kuratowski convergence

In mathematics, Kuratowski convergence is a notion of convergence for sequences (or, more generally, nets) of compact subsets of metric spaces, named after Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".

Definitions

Let (X, d) be a metric space, where X is a set and d is the function of distance between points of X.

For any point x ∈ X and any non-empty compact subset A ⊆ X, define the distance between the point and the subset:

$d(x,A)=\inf\{d(x,a)|a\in A\}$.

For any sequence of such subsets A_n ⊆ X, n ∈ N, the Kuratowski limit inferior (or lower closed limit) of A_n as n → ∞ is

${\mathop {{\mathrm {Li}}}}_{{n\to \infty }}A_{{n}}=\left\{x\in X\left|\limsup _{{n\to \infty }}d(x,A_{{n}})=0\right.\right\}$

$=\left\{x\in X\left|{\begin{matrix}{\mbox{for all open neighbourhoods }}U{\mbox{ of }}x,\\U\cap A_{{n}}\neq \emptyset {\mbox{ for large enough }}n\end{matrix}}\right.\right\};$

the Kuratowski limit superior (or upper closed limit) of A_n as n → ∞ is

${\mathop {{\mathrm {Ls}}}}_{{n\to \infty }}A_{{n}}=\left\{x\in X\left|\liminf _{{n\to \infty }}d(x,A_{{n}})=0\right.\right\}$

$=\left\{x\in X\left|{\begin{matrix}{\mbox{for all open neighbourhoods }}U{\mbox{ of }}x,\\U\cap A_{{n}}\neq \emptyset {\mbox{ for infinitely many }}n\end{matrix}}\right.\right\}.$

If the Kuratowski limits inferior and superior agree (i.e. are the same subset of X), then their common value is called the Kuratowski limit of the sets A_n as n → ∞ and denoted Lt_n→∞A_n.

The definitions for a general net of compact subsets of X go through mutatis mutandis.

Properties

• Although it may seem counter-intuitive that the Kuratowski limit inferior involves the limit superior of the distances, and vice versa, the nomenclature becomes more obvious when one sees that, for any sequence of sets,

${\mathop {{\mathrm {Li}}}}_{{n\to \infty }}A_{{n}}\subseteq {\mathop {{\mathrm {Ls}}}}_{{n\to \infty }}A_{{n}}.$

I.e. the limit inferior is the smaller set and the limit superior the larger one.

• The terms upper and lower closed limit stem from the fact that Li_n→∞A_n and Ls_n→∞A_n are always closed sets in the metric topology on (X, d).

Related Concepts

For metric spaces X we have the following:

• Kuratowski convergence coincides with convergence in Fell topology.
• Kuratowski convergence is weaker than convergence in Vietoris topology.
• Kuratowski convergence is weaker than convergence in Hausdorff metric.
• For compact metric spaces X, Kuratowski convergence coincides with both convergence in Hausdorff metric and Vietoris topology.

Examples

• Let A_n be the zero set of sin(nx) as a function of x from R to itself

$A_{{n}}={\big \{}x\in {\mathbf {R}}{\big |}\sin(nx)=0{\big \}}.$

Then A_n converges in the Kuratowski sense to the whole real line R. Observe that in this case, the A_n do not need to be compact.

See also

Borel–Cantelli lemma

References

• Kuratowski, Kazimierz (1966). Topology. Volumes I and II. New edition, revised and augmented. Translated from the French by J. Jaworowski. New York: Academic Press. pp. xx+560. MR 0217751