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".
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:
For any sequence of such subsets An ⊆ X, n ∈ N, the Kuratowski limit inferior (or lower closed limit) of An as n → ∞ is
the Kuratowski limit superior (or upper closed limit) of An as n → ∞ is
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.
- 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,
- 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 (X, d).
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.
- Let An be the zero set of sin(nx) as a function of x from R to itself
- 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.