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,This is reported in the Commentary section of Chapter 4 of Rockafellar and Wets' text. 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:

:$d(x, A) := \inf_ d(x, y), \qquad x \in X.$

For any sequence of subsets $\_^$ of $X$, the ''Kuratowski limit inferior'' (or ''lower closed limit'') of $A_n$ as $n \to \infty$; is$\begin \mathop A_ :=& \left\ \\ =&\left\; \end$the ''Kuratowski limit superior'' (or ''upper closed limit'') of $A_n$ as $n \to \infty$; is$\begin \mathop A_ :=& \left\ \\ =&\left\; \end$If the Kuratowski limits inferior and superior agree, then the common set is called the ''Kuratowski limit'' of $A_n$ and is denoted $\mathop_ A_n$.

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 $\mathcal(x)$ the collection of open neighbhorhoods of $x$. The ''Kuratowski limit inferior'' of $\_$ is the set$\mathop A_i := \left\,$and the ''Kuratowski limit superior'' is the set$\mathop A_i := \left\.$Elements of $\mathop A_i$ are called ''limit points'' of $\_$ and elements of $\mathop A_i$ 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 $\mathop A_i$.

Examples

* Suppose $(X, d)$ is separable where $X$ is a perfect set, and let $D = \$ be an enumeration of a countable dense subset of $X$. Then the sequence $\_^$ defined by $A_n := \$ has $\mathop A_n = X$.
* Given two closed subsets $B, C \subset X$, defining $A_ := B$ and $A_ := C$ for each $n=1,2,\dots$ yields $\mathop A_n = B \cap C$ and $\mathop A_n = B \cup C$.
* The sequence of closed balls $A_n := \$converges in the sense of Kuratowski when $x_n \to x$ in $X$ and $r_n \to r$ in $[0, +\infty)$, and in particular, $\mathop(A_n) = \$. If $r_n \to +\infty$, then $\mathop A_n = X$ while $\mathop(X \setminus A_n) = \emptyset$.
* Let $A_ := \$. Then $A_n$ converges in the Kuratowski sense to the entire line.
* In a topological vector space, if $\_^$ is a sequence of Convex cone, cones, then so are the Kuratowski limits superior and inferior. For example, the sets $A_n := \$ 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 $\mathop A_n$ and $\mathop A_n$ are closed subsets of $X$, and $\mathop A_n \subset \mathop A_n$ always holds.
* The upper and lower limits do not distinguish between sets and their closures: $\mathop A_n = \mathop \mathop(A_n)$ and $\mathop A_n = \mathop \mathop(A_n)$.
* If $A_n := A$ is a constant sequence, then $\mathop A_n = \mathop A$.
* If $A_n := \$ is a sequence of singletons, then $\mathop A_n$ and $\mathop A_n$ consist of the limit points and cluster points, respectively, of the sequence $\_^ \subset X$.
* If $A_n \subset B_n \subset C_n$ and $B := \mathop A_n = \mathop C_n$, then $\mathop B_n = B$.
* (''Hit and miss criteria'') For a closed subset $A \subset X$, one has
** $A \subset \mathop A_n$, 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 $n_0 \leq n$,
** $\mathop A_n \subset A$, 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 $A_1 \subset A_2 \subset A_3 \subset \cdots$ then the Kuratowski limit exists, and $\mathop A_n = \mathop \left( \bigcup_^ A_n \right)$. Conversely, if $A_1 \supset A_2 \supset A_3 \supset \cdots$ then the Kuratowski limit exists, and $\mathop A_n = \bigcap_^ \mathop(A_n)$.
*If $d_H$ denotes Hausdorff metric, then $d_H(A_n, A) \to 0$ implies $\mathopA = \mathop A_n$. However, noncompact closed sets may converge in the sense of Kuratowski while $d_H(A_n, \mathop A_n) = +\infty$ for each $n=1,2,\dots$
*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 $S^(y) = \$. We can define the operators$\begin \mathop_ S(x') :=& \bigcap_ \mathop S(x'), \qquad x \in X \\ \mathop_ S(x') :=& \bigcup_ \mathop S(x'), \qquad x \in X\\ \end$where $x' \to x$ means convergence in sequences when $X$ is metrizable and convergence in nets otherwise. Then,

* $S$ is ''inner semi-continuous'' at $x \in X$ if $S(x) \subset \mathop_ S(x')$;
* $S$ is ''outer semi-continuous'' at $x \in X$ if $\mathop_ S(x') \subset S(x)$.

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.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 f 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. 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) = \$ is continuous X \times ,+\infty) \rightrightarrows X.
* Given a function f : X \to [-\infty, +\infty/math>, the superlevel set mapping $S_f(x) := \$ 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 $V \in \mathcal(y)$ and $U \in \mathcal(x)$ such that $U \cap S^(V) = \emptyset$.
* $S$ is inner semi-continuous at $x$, if and only if for every $y \in S(x)$ and neighborhood $V \in \mathcal(y)$ there is a neighborhood $U \in \mathcal(x)$ 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 $\$ 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 hemicontinuous">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 $S : \mathbb^n \to \mathbb^m$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 $T(x) := \mathop S(x)$ is also inner semi-continuous.

Epi-convergence and Γ-convergence

For the metric space $(X, d)$ a sequence of functions $f_n : X \to [-\infty, +\infty]$, the ''epi-limit inferior'' (or ''lower epi-limit'') is the function $\mathop f_n$ defined by the Epigraph (mathematics), epigraph equation$\mathop \left( \mathop f_n\right) := \mathop \left(\mathop f_n\right),$and similarly the ''epi-limit superior'' (or ''upper epi-limit'') is the function $\mathop f_n$ defined by the epigraph equation$\mathop \left( \mathop f_n\right) := \mathop \left(\mathop f_n\right).$Since Kuratowski upper and lower limits are closed sets, it follows that both $\mathop f_n$ and $\mathop f_n$ are lower semi-continuous functions. Similarly, since $\mathop \mathop f_n \subset \mathop \mathop f_n$, it follows that $\mathop f_n \leq \mathop f_n$ uniformly. These functions agree, if and only if $\mathop \mathop f_n$ 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.

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

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* {{cite book , last1=Rockafellar , first1=R. Tyrrell , url=https://www.worldcat.org/oclc/883392544 , title=Variational analysis , last2=Wets , first2=Roger J.-B. , date=1998 , isbn=978-3-642-02431-3 , edition= , location=Berlin , oclc=883392544

Metric geometry