# 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 (Xd) 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:

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

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

${\displaystyle {\mathop {\mathrm {Li} }}_{n\to \infty }A_{n}=\left\{x\in X\left|\limsup _{n\to \infty }d(x,A_{n})=0\right.\right\}}$
${\displaystyle =\left\{x\in X\left|{\begin{matrix}{\mbox{for all open neighbourhoods }}U{\mbox{ of }}x,\\U\cap A_{n}\neq \emptyset {\mbox{ for large enough }}n\end{matrix}}\right.\right\};}$

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

${\displaystyle {\mathop {\mathrm {Ls} }}_{n\to \infty }A_{n}=\left\{x\in X\left|\liminf _{n\to \infty }d(x,A_{n})=0\right.\right\}}$
${\displaystyle =\left\{x\in X\left|{\begin{matrix}{\mbox{for all open neighbourhoods }}U{\mbox{ of }}x,\\U\cap A_{n}\neq \emptyset {\mbox{ for infinitely many }}n\end{matrix}}\right.\right\}.}$

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 An as n → ∞ and denoted Ltn→∞An.

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,
${\displaystyle {\mathop {\mathrm {Li} }}_{n\to \infty }A_{n}\subseteq {\mathop {\mathrm {Ls} }}_{n\to \infty }A_{n}.}$
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 Lin→∞An and Lsn→∞An are always closed sets in the metric topology on (Xd).

## 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 An be the zero set of sin(nx) as a function of x from R to itself
${\displaystyle A_{n}={\big \{}x\in \mathbf {R} {\big |}\sin(nx)=0{\big \}}.}$
Then An converges in the Kuratowski sense to the whole real line R. Observe that in this case, the An do not need to be compact.