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 the Polish mathematician Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".

Definitions

Let (X, d) be a metric space. For any point x ∈ X and any non-empty compact subset A ⊆ X, let

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

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 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).

Examples

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

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

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.

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. MR0217751