In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902, the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".

Definitions

For a given sequence of points in a space X, a limit point of the sequence can be understood as any point x \in X where the sequence eventually becomes arbitrarily close to x. On the other hand, a cluster point of the sequence can be thought of as a point x \in X where the sequence frequently becomes arbitrarily close to x. The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space X.

Metric Spaces

Let (X,d) be a metric space, where X is a given set. For any point x and any non-empty subset A \subset X, define the distance between the point and the subset: For any sequence of subsets of X, the Kuratowski limit inferior (or lower closed limit) of A_n as ; isthe Kuratowski limit superior (or upper closed limit) of A_n as ; isIf the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of A_n and is denoted.

Topological Spaces

If (X, \tau) is a topological space, and are a net of subsets of X, the limits inferior and superior follow a similar construction. For a given point x \in X denote the collection of open neighborhoods of x. The Kuratowski limit inferior of is the setand the Kuratowski limit superior is the setElements of are called limit points of and elements of are called cluster points of. In other words, x is a limit point of if each of its neighborhoods intersects A_i for all i in a "residual" subset of I, while x is a cluster point of if each of its neighborhoods intersects A_i for all i in a cofinal subset of I. When these sets agree, the common set is the Kuratowski limit of, denoted.

Examples

Properties

The following properties hold for the limits inferior and superior in both the metric and topological contexts, but are stated in the metric formulation for ease of reading.

Kuratowski Continuity of Set-Valued Functions

Let be a set-valued function between the spaces X and Y; namely, for all x \in X. Denote. We can define the operatorswhere x' \to x means convergence in sequences when X is metrizable and convergence in nets otherwise. Then, When S is both inner and outer semi-continuous at x \in X, we say that S is continuous (or continuous in the sense of Kuratowski). Continuity of set-valued functions is commonly defined in terms of lower- and upper-hemicontinuity popularized by Berge. In this sense, a set-valued function is continuous if and only if the function defined by f(x) = S(x) is continuous with respect to the Vietoris hyperspace topology of 2^Y. For set-valued functions with closed values, continuity in the sense of Vietoris-Berge is stronger than continuity in the sense of Kuratowski.

Examples

Properties

Epi-convergence and Γ-convergence

For the metric space (X, d) a sequence of functions, the epi-limit inferior (or lower epi-limit) is the function defined by the epigraph equationand similarly the epi-limit superior (or upper epi-limit) is the function defined by the epigraph equationSince Kuratowski upper and lower limits are closed sets, it follows that both and are lower semi-continuous functions. Similarly, since, it follows that uniformly. These functions agree, if and only if exists, and the associated function is called the epi-limit of. When (X, \tau) is a topological space, epi-convergence of the sequence is called Γ-convergence. From the perspective of Kuratowski convergence there is no distinction between epi-limits and Γ-limits. The concepts are usually studied separately, because epi-convergence admits special characterizations that rely on the metric space structure of X, which does not hold in topological spaces generally.