Kuratowski convergence

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

Metric Spaces

Let be a metric space, where is a given set. For any point and any non-empty subset , define the distance between the point and the subset:

For any sequence of subsets of , the Kuratowski limit inferior (or lower closed limit) of as ; isthe Kuratowski limit superior (or upper closed limit) of as ; isIf the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of and is denoted .

Topological Spaces

If is a topological space, and are a net of subsets of , the limits inferior and superior follow a similar construction. For a given point denote the collection of open neighborhoods of . 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, is a limit point of if each of its neighborhoods intersects for all in a "residual" subset of , while is a cluster point of if each of its neighborhoods intersects for all in a cofinal subset of .

When these sets agree, the common set is the Kuratowski limit of , denoted .

Examples

• Suppose is separable where is a perfect set, and let be an enumeration of a countable dense subset of . Then the sequence defined by has .
• Given two closed subsets , defining and for each yields and .
• The sequence of closed balls converges in the sense of Kuratowski when in and in , and in particular, . If , then while .
• Let . Then converges in the Kuratowski sense to the entire line.
• In a topological vector space, if is a sequence of cones, then so are the Kuratowski limits superior and inferior. For example, the sets converge to .

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.

• Both and are closed subsets of , and always holds.
• The upper and lower limits do not distinguish between sets and their closures: and .
• If is a constant sequence, then .
• If is a sequence of singletons, then and consist of the limit points and cluster points, respectively, of the sequence .
• If and , then .
• (Hit and miss criteria) For a closed subset , one has
• , if and only if for every open set with there exists such that for all ,
• , if and only if for every compact set with there exists such that for all .
• If then the Kuratowski limit exists, and . Conversely, if then the Kuratowski limit exists, and .
• If denotes Hausdorff metric, then implies . However, noncompact closed sets may converge in the sense of Kuratowski while for each
• Convergence in the sense of Kuratowski is weaker than convergence in the sense of Vietoris but equivalent to convergence in the sense of Fell. If is compact, then these are all equivalent and agree with convergence in Hausdorff metric.

Kuratowski Continuity of Set-Valued Functions

Let be a set-valued function between the spaces and ; namely, for all . Denote . We can define the operatorswhere means convergence in sequences when is metrizable and convergence in nets otherwise. Then,

• is inner semi-continuous at if ;
• is outer semi-continuous at if .

When is both inner and outer semi-continuous at , we say that 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 is continuous with respect to the Vietoris hyperspace topology of . For set-valued functions with closed values, continuity in the sense of Vietoris-Berge is stronger than continuity in the sense of Kuratowski.

Examples

• The set-valued function is continuous .
• Given a function , the superlevel set mapping is outer semi-continuous at , if and only if is lower semi-continuous at . Similarly, is inner semi-continuous at , if and only if is upper semi-continuous at .

Properties

• If is continuous at , then is closed.
• is outer semi-continuous at , if and only if for every there are neighborhoods and such that .
• is inner semi-continuous at , if and only if for every and neighborhood there is a neighborhood such that for all .
• is (globally) outer semi-continuous, if and only if its graph is closed.
• (Relations to Vietoris-Berge continuity). Suppose is closed.
• is inner semi-continuous at , if and only if is lower hemi-continuous at in the sense of Vietoris-Berge.
• If is upper hemi-continuous at , then is outer semi-continuous at . The converse is false in general, but holds when is a compact space.
• If has a convex graph, then is inner semi-continuous at each point of the interior of the domain of . Conversely, given any inner semi-continuous set-valued function , the convex hull mapping is also inner semi-continuous.

Epi-convergence and Γ-convergence

For the metric space 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 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 , which does not hold in topological spaces generally.

See also

• Set-theoretic limit
• Borel–Cantelli lemma, but note that set-theoretic limits and Kuratowski limits do not agree.
• Wijsman convergence
• Hausdorff distance
• Hemicontinuity
• Vietoris topology
• Epi-convergence
• Gamma convergence

Notes

References