Kuratowski convergence explained

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,^[1] the concept was popularized in texts by Felix Hausdorff^[2] and Kazimierz Kuratowski.^[3] Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".

Definitions

For a given sequence

\{x _{n\}} ^{inf ty} ^{n =1} 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:d(x, A) : = inf _{y} d(x, y), x \in X.

For any sequence of subsets

\{A _{n\}} ^{inf ty} ^{n =1} of X , the Kuratowski limit inferior (or lower closed limit) of A _{n} as n \to inf ty ; isthe Kuratowski limit superior (or upper closed limit) of A _{n} as n \to inf ty ; isIf the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of A _{n} and is denoted Lim _{n} A _{n} .

Topological Spaces

If $(X,\; \backslash tau)$ is a topological space, and $\backslash \_$ are a net of subsets of $X$, the limits inferior and superior follow a similar construction. For a given point $x\; \backslash in\; X$ denote $\backslash mathcal(x)$ the collection of open neighborhoods of $x$. The Kuratowski limit inferior of $\backslash \_$ is the set$$\backslash mathop\; A\_i\; :=\; \backslash left\backslash ,$$and the Kuratowski limit superior is the set$$\backslash mathop\; A\_i\; :=\; \backslash left\backslash .$$Elements of $\backslash mathop\; A\_i$ are called limit points of $\backslash \_$ and elements of $\backslash mathop\; A\_i$ are called cluster points of $\backslash \_$. In other words,

x is a limit point of $\backslash \_$ if each of its neighborhoods intersects A _{i} for all i in a "residual" subset of I , while x is a cluster point of $\backslash \_$ 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 $\backslash \_$, denoted

Lim A _{i} .

Examples

• Suppose

(X, d) is separable where X is a perfect set, and let D = \{d _{1,} d _{2,} ... \} be an enumeration of a countable dense subset of X . Then the sequence \{A _{n\}} ^{inf ty} ^{n =1} defined by A _{n} : = \{d _{1,} d _{2,} ... , d _{n\}} has Lim A _{n} = X .

• Given two closed subsets

B, C \subset X , defining A _{2n-1} : = B and A _{2n} : = C for each n =1 , 2 , ... yields Li A _{n} = B \cap C and Ls A _{n} = B \cup C .A _{n} : = \{y \in X: d(x _{n,y)} \leq r _{n\}} converges in the sense of Kuratowski when x _{n} \to x in X and r _{n} \to r in [ 0 , +inf ty) , and in particular, Lim (A _{n)} = \{y \in X : d(x,y) \leq r\} . If r _{n} \to +inf ty , then Lim A _{n} = X while Lim (X \setminus A _{n)} = \emptyset .

• Let $A\_\; :=\; \backslash$. Then

A _{n} converges in the Kuratowski sense to the entire line.

• In a topological vector space, if

\{A _{n\}} ^{inf ty} ^{n =1} is a sequence of cones, then so are the Kuratowski limits superior and inferior. For example, the sets A _{n} : = \{(x,y) \in R ^{2} : y \geq n|x|\} converge to \{( 0 ,y) \in R ^{2} : y \geq 0 \} .

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.^[4]

• Both

Li A _{n} and Ls A _{n} are closed subsets of X , and Li A _{n} \subset Ls A _{n} always holds.

• The upper and lower limits do not distinguish between sets and their closures:

Li A _{n} = Li cl (A _{n)} and Ls A _{n} = Ls cl (A _{n)} .

• If

A _{n} : = A is a constant sequence, then Lim A _{n} = cl A .

• If

A _{n} : = \{x _{n\}} is a sequence of singletons, then Li A _{n} and Ls A _{n} consist of the limit points and cluster points, respectively, of the sequence \{x _{n\}} ^{inf ty} ^{n =1} \subset X .

• If

A _{n} \subset B _{n} \subset C _{n} and B : = Lim A _{n} = Lim C _{n} , then Lim B _{n} = B .

• (Hit and miss criteria) For a closed subset

A \subset X , one hasA \subset Li A _{n} , if and only if for every open set U \subset X with A \cap U \ne \emptyset there exists n _{0} such that A _{n} \cap U \ne \emptyset for all n _{0} \leq n ,Ls A _{n} \subset A , if and only if for every compact set K \subset X with A \cap K \ne \emptyset there exists n _{0} such that A _{n} \cap K \ne \emptyset for all n _{0} \leq n .

• If

A _{1} \subset A _{2} \subset A _{3} \subset … then the Kuratowski limit exists, and $\backslash mathop\; A\_n\; =\; \backslash mathop\; \backslash left(\backslash bigcup\_^\; A\_n\; \backslash right)$. Conversely, if A _{1} \supset A _{2} \supset A _{3} \supset … then the Kuratowski limit exists, and $\backslash mathop\; A\_n\; =\; \backslash bigcap\_^\; \backslash mathop(A\_n)$.

• If

d _{H} denotes Hausdorff metric, then d _{H(A} _{n,} A) \to 0 implies cl A = Lim A _{n} . However, noncompact closed sets may converge in the sense of Kuratowski while d _{H(A} _{n,} Lim A _{n)} = +inf ty for each n =1 , 2 , ... ^[5]

• 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

X is compact, then these are all equivalent and agree with convergence in Hausdorff metric.

Kuratowski Continuity of Set-Valued Functions

Let

S : X \rightrightarrows Y be a set-valued function between the spaces X and Y ; namely, S(x) \subset Y for all x \in X . Denote S ^{-1} (y) = \{x \in X : y \in S(x)\} . We can define the operatorswhere x' \to x means convergence in sequences when X is metrizable and convergence in nets otherwise. Then,S is inner semi-continuous at x \in X if $S(x)\; \backslash subset\; \backslash mathop\_\; S(x\text{'})$;S is outer semi-continuous at x \in X if $\backslash mathop\_\; S(x\text{'})\; \backslash subset\; S(x)$.

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.^[6] In this sense, a set-valued function is continuous if and only if the function

f _{S} : X \to 2 ^{Y} 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

• The set-valued function

B(x,r) = \{y \in X : d(x,y) \leq r \} is continuous X x [ 0 , +inf ty) \rightrightarrows X .

• Given a function

f : X \to [ -inf ty, +inf ty] , the superlevel set mapping S _{f(x)} : = \{ λ \in R : f(x) \leq λ \} is outer semi-continuous at x , if and only if f is lower semi-continuous at x . Similarly, S _{f} is inner semi-continuous at x , if and only if f is upper semi-continuous at x .

Properties

• If

S is continuous at x , then S(x) is closed.S is outer semi-continuous at x , if and only if for every y \notin S(x) there are neighborhoods V \in l{N}(y) and U \in l{N}(x) such that U \cap S ^{-1} (V) = \emptyset .S is inner semi-continuous at x , if and only if for every y \in S(x) and neighborhood V \in l{N}(y) there is a neighborhood U \in l{N}(x) such that V \cap S(x') \ne \emptyset for all x' \in U .S is (globally) outer semi-continuous, if and only if its graph \{(x,y) \in X x Y : y \in S(x)\} is closed.

• (Relations to Vietoris-Berge continuity). Suppose

S(x) is closed.S is inner semi-continuous at x , if and only if S is lower hemi-continuous at x in the sense of Vietoris-Berge.

• • If

S is upper hemi-continuous at x , then S is outer semi-continuous at x . The converse is false in general, but holds when Y is a compact space.

• If

S : R ^{n} \to R ^{m} has a convex graph, then S is inner semi-continuous at each point of the interior of the domain of S . Conversely, given any inner semi-continuous set-valued function S , the convex hull mapping T(x) : = conv S(x) is also inner semi-continuous.

Epi-convergence and Γ-convergence

See main article: Epi-convergence and Γ-convergence. For the metric space

(X, d) a sequence of functions f _{n} : X \to [ -inf ty, +inf ty] , the epi-limit inferior (or lower epi-limit) is the function e \liminf f _{n} defined by the epigraph equation$$\backslash mathop\; \backslash left(\backslash mathop\; f\_n\backslash right)\; :=\; \backslash mathop\; \backslash left(\backslash mathop\; f\_n\backslash right),$$and similarly the epi-limit superior (or upper epi-limit) is the function e \limsup f _{n} defined by the epigraph equation$$\backslash mathop\; \backslash left(\backslash mathop\; f\_n\backslash right)$$

= \mathop \left(\mathop f_n\right).Since Kuratowski upper and lower limits are closed sets, it follows that both

e \liminf f _{n} and e \limsup f _{n} are lower semi-continuous functions. Similarly, since Li epi f _{n} \subset Ls epi f _{n} , it follows that e \liminf f _{n} \leq e \liminf f _{n} uniformly. These functions agree, if and only if Lim epi f _{n} exists, and the associated function is called the epi-limit of \{f _{n\}} ^{inf ty} ^{n =1} .

When

(X, \tau) is a topological space, epi-convergence of the sequence \{f _{n\}} ^{inf ty} ^{n =1} 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.

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

1. This is reported in the Commentary section of Chapter 4 of Rockafellar and Wets' text.
2. Book: Hausdorff, Felix . Mengenlehre . . 1927 . 2nd . Berlin . de.
3. Book: Kuratowski, Kazimierz . Topologie, I & II . Panstowowe Wyd Nauk . 1933 . Warsaw . fr.
4. The interested reader may consult Beer's text, in particular Chapter 5, Section 2, for these and more technical results in the topological setting. For Euclidean spaces, Rockafellar and Wets report similar facts in Chapter 4.
5. For an example, consider the sequence of cones in the previous section.
6. Rockafellar and Wets write in the Commentary to Chapter 6 of their text: "The terminology of 'inner' and 'outer' semicontinuity, instead of 'lower' and 'upper', has been foorced on us by the fact that the prevailing definition of 'upper semicontinuity' in the literature is out of step with developments in set convergence and the scope of applications that must be handled, now that mappings S with unbounded range and even unbounded value sets S(x) are so important... Despite the historical justification, the tide can no longer be turned in the meaning of 'upper semicontinuity', yet the concept of 'continuity' is too crucial for applications to be left in the poorly usable form that rests on such an unfortunately restrictive property [of upper semicontinuity]"; see pages 192-193. Note also that authors differ on whether "semi-continuity" or "hemi-continuity" is the preferred language for Vietoris-Berge continuity concepts.

References

• Book: Beer , Gerald . Topologies on closed and closed convex sets. Mathematics and its Applications. Kluwer Academic Publishers Group. Dordrecht. 1993. xii+340.
• Book: Kuratowski, Kazimierz <!--. Kazimierz Kuratowski-->

. Kazimierz Kuratowski--> . Topology. Volumes I and II . New edition, revised and augmented. Translated from the French by J. Jaworowski . Academic Press . New York . 1966 . xx+560.

• Book: Rockafellar . R. Tyrrell . Variational analysis . Wets . Roger J.-B. . 1998 . 978-3-642-02431-3 . Berlin . 883392544.