# 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:

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

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

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

the Kuratowski limit superior (or upper closed limit) of An 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 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,
$\mathop{\mathrm{Li}}_{n \to \infty} A_{n} \subseteq \mathop{\mathrm{Ls}}_{n \to \infty} A_{n}.$
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
$A_{n} = \big\{ x \in \mathbf{R} \big| \sin (n x) = 0 \big\}.$
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.