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

:$d(x,\; A)\; =\; inf\_\{a\; in\; A\}\; d(x,\; a).$

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

:$mathop\{mathrm\{Li\_\{n\; o\; infty\}\; A\_\{n\}\; =\; left\{\; x\; in\; X\; left|\; limsup\_\{n\; o\; infty\}\; d(x,\; A\_\{n\})\; =\; 0\; ight.\; ight\}$::

the Kuratowski limit superior (or upper closed limit) of "A"_"n" as "n" → ∞ is

:$mathop\{mathrm\{Ls\_\{n\; o\; infty\}\; A\_\{n\}\; =\; left\{\; x\; in\; X\; left|\; liminf\_\{n\; o\; infty\}\; d(x,\; A\_\{n\})\; =\; 0\; ight.\; ight\}$::

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,

::$mathop\{mathrm\{Li\_\{n\; o\; infty\}\; A\_\{n\}\; subseteq\; mathop\{mathrm\{Ls\_\{n\; o\; infty\}\; A\_\{n\}.$

: 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

::

: Then "A"_"n" converges in the Kuratowski sense to the whole real line R.

References

* cite book
last = Kuratowski
first = Kazimierz
authorlink = Kazimierz Kuratowski
title = Topology. Volumes I and II
series = New edition, revised and augmented. Translated from the French by J. Jaworowski
publisher = Academic Press
location = New York
year = 1966
pages = xx+560 MathSciNet|id=0217751