Kuratowski convergence


In mathematics, Kuratowski convergence is a notion of convergence for sequences 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 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 of An as n → ∞ is
the Kuratowski limit superior of An as n → ∞ is
If the Kuratowski limits inferior and superior agree, 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

For metric spaces X we have the following: