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,^[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 ${\displaystyle \{x_{n}\}_{n=1}^{\infty }}$ of points in a space ${\displaystyle X}$, a limit point of the sequence can be understood as any point ${\displaystyle x\in X}$ where the sequence eventually becomes arbitrarily close to ${\displaystyle x}$. On the other hand, a cluster point of the sequence can be thought of as a point ${\displaystyle x\in X}$ where the sequence frequently becomes arbitrarily close to ${\displaystyle x}$. The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space ${\displaystyle X}$.

Metric Spaces

Let ${\displaystyle (X,d)}$ be a metric space, where ${\displaystyle X}$ is a given set. For any point ${\displaystyle x}$ and any non-empty subset ${\displaystyle A\subset X}$, define the distance between the point and the subset:

${\displaystyle d(x,A):=\inf _{y\in A}d(x,y),\qquad x\in X.}$

For any sequence of subsets ${\displaystyle \{A_{n}\}_{n=1}^{\infty }}$ of ${\displaystyle X}$, the Kuratowski limit inferior (or lower closed limit) of ${\displaystyle A_{n}}$ as ${\displaystyle n\to \infty }$; is

the Kuratowski limit superior (or upper closed limit) of ${\displaystyle A_{n}}$ as ${\displaystyle n\to \infty }$; isIf the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of ${\displaystyle A_{n}}$ and is denoted ${\displaystyle \mathop {\mathrm {Lim} } _{n\to \infty }A_{n}}$.

Topological Spaces

If ${\textstyle (X,\tau )}$ is a topological space, and ${\textstyle \{A_{i}\}_{i\in I}}$ are a net of subsets of ${\textstyle X}$, the limits inferior and superior follow a similar construction. For a given point ${\textstyle x\in X}$ denote ${\textstyle {\mathcal {N}}(x)}$ the collection of open neighborhoods of ${\textstyle x}$. The Kuratowski limit inferior of ${\textstyle \{A_{i}\}_{i\in I}}$ is the set

and the Kuratowski limit superior is the setElements of ${\textstyle \mathop {\mathrm {Li} } A_{i}}$ are called limit points of ${\textstyle \{A_{i}\}_{i\in I}}$ and elements of ${\textstyle \mathop {\mathrm {Ls} } A_{i}}$ are called cluster points of ${\textstyle \{A_{i}\}_{i\in I}}$. In other words, ${\displaystyle x}$ is a limit point of ${\textstyle \{A_{i}\}_{i\in I}}$ if each of its neighborhoods intersects ${\displaystyle A_{i}}$ for all ${\displaystyle i}$ in a "residual" subset of ${\displaystyle I}$, while ${\displaystyle x}$ is a cluster point of ${\textstyle \{A_{i}\}_{i\in I}}$ if each of its neighborhoods intersects ${\displaystyle A_{i}}$ for all ${\displaystyle i}$ in a cofinal subset of ${\displaystyle I}$.

When these sets agree, the common set is the Kuratowski limit of ${\textstyle \{A_{i}\}_{i\in I}}$, denoted ${\displaystyle \mathop {\mathrm {Lim} } A_{i}}$.

Examples

• Suppose ${\displaystyle (X,d)}$ is separable where ${\displaystyle X}$ is a perfect set, and let ${\displaystyle D=\{d_{1},d_{2},\dots \}}$ be an enumeration of a countable dense subset of ${\displaystyle X}$. Then the sequence ${\displaystyle \{A_{n}\}_{n=1}^{\infty }}$ defined by ${\displaystyle A_{n}:=\{d_{1},d_{2},\dots ,d_{n}\}}$ has ${\displaystyle \mathop {\mathrm {Lim} } A_{n}=X}$.
• Given two closed subsets ${\displaystyle B,C\subset X}$, defining ${\displaystyle A_{2n-1}:=B}$ and ${\displaystyle A_{2n}:=C}$ for each ${\displaystyle n=1,2,\dots }$ yields ${\displaystyle \mathop {\mathrm {Li} } A_{n}=B\cap C}$ and ${\displaystyle \mathop {\mathrm {Ls} } A_{n}=B\cup C}$.
• The sequence of closed balls ${\displaystyle A_{n}:=\{y\in X:d(x_{n},y)\leq r_{n}\}}$converges in the sense of Kuratowski when ${\displaystyle x_{n}\to x}$ in ${\displaystyle X}$ and ${\displaystyle r_{n}\to r}$ in ${\displaystyle [0,+\infty )}$, and in particular, ${\displaystyle \mathop {\mathrm {Lim} } (A_{n})=\{y\in X:d(x,y)\leq r\}}$. If ${\displaystyle r_{n}\to +\infty }$, then ${\displaystyle \mathop {\mathrm {Lim} } A_{n}=X}$ while ${\displaystyle \mathop {\mathrm {Lim} } (X\setminus A_{n})=\emptyset }$.
• Let ${\textstyle A_{n}:=\{x\in \mathbb {R} :\sin(nx)=0\}}$. Then ${\displaystyle A_{n}}$ converges in the Kuratowski sense to the entire line.
• In a topological vector space, if ${\displaystyle \{A_{n}\}_{n=1}^{\infty }}$ is a sequence of cones, then so are the Kuratowski limits superior and inferior. For example, the sets ${\displaystyle A_{n}:=\{(x,y)\in \mathbb {R} ^{2}:y\geq n|x|\}}$ converge to ${\displaystyle \{(0,y)\in \mathbb {R} ^{2}:y\geq 0\}}$.

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 ${\displaystyle \mathop {\mathrm {Li} } A_{n}}$ and ${\displaystyle \mathop {\mathrm {Ls} } A_{n}}$ are closed subsets of ${\displaystyle X}$, and ${\displaystyle \mathop {\mathrm {Li} } A_{n}\subset \mathop {\mathrm {Ls} } A_{n}}$ always holds.
• The upper and lower limits do not distinguish between sets and their closures: ${\displaystyle \mathop {\mathrm {Li} } A_{n}=\mathop {\mathrm {Li} } \mathop {\mathrm {cl} } (A_{n})}$ and ${\displaystyle \mathop {\mathrm {Ls} } A_{n}=\mathop {\mathrm {Ls} } \mathop {\mathrm {cl} } (A_{n})}$.
• If ${\displaystyle A_{n}:=A}$ is a constant sequence, then ${\displaystyle \mathop {\mathrm {Lim} } A_{n}=\mathop {\mathrm {cl} } A}$.
• If ${\displaystyle A_{n}:=\{x_{n}\}}$ is a sequence of singletons, then ${\displaystyle \mathop {\mathrm {Li} } A_{n}}$ and ${\displaystyle \mathop {\mathrm {Ls} } A_{n}}$ consist of the limit points and cluster points, respectively, of the sequence ${\displaystyle \{x_{n}\}_{n=1}^{\infty }\subset X}$.
• If ${\displaystyle A_{n}\subset B_{n}\subset C_{n}}$ and ${\displaystyle B:=\mathop {\mathrm {Lim} } A_{n}=\mathop {\mathrm {Lim} } C_{n}}$, then ${\displaystyle \mathop {\mathrm {Lim} } B_{n}=B}$.
• (Hit and miss criteria) For a closed subset ${\displaystyle A\subset X}$, one has
  • ${\displaystyle A\subset \mathop {\mathrm {Li} } A_{n}}$, if and only if for every open set ${\displaystyle U\subset X}$ with ${\displaystyle A\cap U\neq \emptyset }$ there exists ${\displaystyle n_{0}}$ such that ${\displaystyle A_{n}\cap U\neq \emptyset }$ for all ${\displaystyle n_{0}\leq n}$,
  • ${\displaystyle \mathop {\mathrm {Ls} } A_{n}\subset A}$, if and only if for every compact set ${\displaystyle K\subset X}$ with ${\displaystyle A\cap K\neq \emptyset }$ there exists ${\displaystyle n_{0}}$ such that ${\displaystyle A_{n}\cap K\neq \emptyset }$ for all ${\displaystyle n_{0}\leq n}$.
• If ${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \cdots }$ then the Kuratowski limit exists, and ${\textstyle \mathop {\mathrm {Lim} } A_{n}=\mathop {\mathrm {cl} } \left(\bigcup _{n=1}^{\infty }A_{n}\right)}$. Conversely, if ${\displaystyle A_{1}\supset A_{2}\supset A_{3}\supset \cdots }$ then the Kuratowski limit exists, and ${\textstyle \mathop {\mathrm {Lim} } A_{n}=\bigcap _{n=1}^{\infty }\mathop {\mathrm {cl} } (A_{n})}$.
• If ${\displaystyle d_{H}}$ denotes Hausdorff metric, then ${\displaystyle d_{H}(A_{n},A)\to 0}$ implies ${\displaystyle \mathop {\mathrm {cl} } A=\mathop {\mathrm {Lim} } A_{n}}$. However, noncompact closed sets may converge in the sense of Kuratowski while ${\displaystyle d_{H}(A_{n},\mathop {\mathrm {Lim} } A_{n})=+\infty }$ for each ${\displaystyle n=1,2,\dots }$^[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 ${\displaystyle X}$ is compact, then these are all equivalent and agree with convergence in Hausdorff metric.

Kuratowski Continuity of Set-Valued Functions

Let ${\displaystyle S:X\rightrightarrows Y}$ be a set-valued function between the spaces ${\displaystyle X}$ and ${\displaystyle Y}$; namely, ${\displaystyle S(x)\subset Y}$ for all ${\displaystyle x\in X}$. Denote ${\displaystyle S^{-1}(y)=\{x\in X:y\in S(x)\}}$. We can define the operators

where ${\displaystyle x'\to x}$ means convergence in sequences when ${\displaystyle X}$ is metrizable and convergence in nets otherwise. Then,

• ${\displaystyle S}$ is inner semi-continuous at ${\displaystyle x\in X}$ if ${\textstyle S(x)\subset \mathop {\mathrm {Li} } _{x'\to x}S(x')}$;
• ${\displaystyle S}$ is outer semi-continuous at ${\displaystyle x\in X}$ if ${\textstyle \mathop {\mathrm {Ls} } _{x'\to x}S(x')\subset S(x)}$.

When ${\displaystyle S}$ is both inner and outer semi-continuous at ${\displaystyle x\in X}$, we say that ${\displaystyle S}$ 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 ${\displaystyle f_{S}:X\to 2^{Y}}$ defined by ${\displaystyle f(x)=S(x)}$ is continuous with respect to the Vietoris hyperspace topology of ${\displaystyle 2^{Y}}$. 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 ${\displaystyle B(x,r)=\{y\in X:d(x,y)\leq r\}}$ is continuous ${\displaystyle X\times [0,+\infty )\rightrightarrows X}$.
• Given a function ${\displaystyle f:X\to [-\infty ,+\infty ]}$, the superlevel set mapping ${\displaystyle S_{f}(x):=\{\lambda \in \mathbb {R} :f(x)\leq \lambda \}}$ is outer semi-continuous at ${\displaystyle x}$, if and only if ${\displaystyle f}$ is lower semi-continuous at ${\displaystyle x}$. Similarly, ${\displaystyle S_{f}}$ is inner semi-continuous at ${\displaystyle x}$, if and only if ${\displaystyle f}$ is upper semi-continuous at ${\displaystyle x}$.

Properties

• If ${\displaystyle S}$ is continuous at ${\displaystyle x}$, then ${\displaystyle S(x)}$ is closed.
• ${\displaystyle S}$ is outer semi-continuous at ${\displaystyle x}$, if and only if for every ${\displaystyle y\notin S(x)}$ there are neighborhoods ${\displaystyle V\in {\mathcal {N}}(y)}$ and ${\displaystyle U\in {\mathcal {N}}(x)}$ such that ${\displaystyle U\cap S^{-1}(V)=\emptyset }$.
• ${\displaystyle S}$ is inner semi-continuous at ${\displaystyle x}$, if and only if for every ${\displaystyle y\in S(x)}$ and neighborhood ${\displaystyle V\in {\mathcal {N}}(y)}$ there is a neighborhood ${\displaystyle U\in {\mathcal {N}}(x)}$ such that ${\displaystyle V\cap S(x')\neq \emptyset }$ for all ${\displaystyle x'\in U}$.
• ${\displaystyle S}$ is (globally) outer semi-continuous, if and only if its graph ${\displaystyle \{(x,y)\in X\times Y:y\in S(x)\}}$ is closed.
• (Relations to Vietoris-Berge continuity). Suppose ${\displaystyle S(x)}$ is closed.
  • ${\displaystyle S}$ is inner semi-continuous at ${\displaystyle x}$, if and only if ${\displaystyle S}$ is lower hemi-continuous at ${\displaystyle x}$ in the sense of Vietoris-Berge.
  • If ${\displaystyle S}$ is upper hemi-continuous at ${\displaystyle x}$, then ${\displaystyle S}$ is outer semi-continuous at ${\displaystyle x}$. The converse is false in general, but holds when ${\displaystyle Y}$ is a compact space.
• If ${\displaystyle S:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$has a convex graph, then ${\displaystyle S}$ is inner semi-continuous at each point of the interior of the domain of ${\displaystyle S}$. Conversely, given any inner semi-continuous set-valued function ${\displaystyle S}$, the convex hull mapping ${\displaystyle T(x):=\mathop {\mathrm {conv} } S(x)}$ is also inner semi-continuous.

Epi-convergence and Γ-convergence

For the metric space ${\displaystyle (X,d)}$ a sequence of functions ${\displaystyle f_{n}:X\to [-\infty ,+\infty ]}$, the epi-limit inferior (or lower epi-limit) is the function ${\displaystyle \mathop {\mathrm {e} \liminf } f_{n}}$ defined by the epigraph equation

and similarly the epi-limit superior (or upper epi-limit) is the function ${\displaystyle \mathop {\mathrm {e} \limsup } f_{n}}$ defined by the epigraph equationSince Kuratowski upper and lower limits are closed sets, it follows that both ${\displaystyle \mathop {\mathrm {e} \liminf } f_{n}}$ and ${\displaystyle \mathop {\mathrm {e} \limsup } f_{n}}$ are lower semi-continuous functions. Similarly, since ${\displaystyle \mathop {\mathrm {Li} } \mathop {\mathrm {epi} } f_{n}\subset \mathop {\mathrm {Ls} } \mathop {\mathrm {epi} } f_{n}}$, it follows that ${\displaystyle \mathop {\mathrm {e} \liminf } f_{n}\leq \mathop {\mathrm {e} \liminf } f_{n}}$ uniformly. These functions agree, if and only if ${\displaystyle \mathop {\mathrm {Lim} } \mathop {\mathrm {epi} } f_{n}}$ exists, and the associated function is called the epi-limit of ${\displaystyle \{f_{n}\}_{n=1}^{\infty }}$.

When ${\displaystyle (X,\tau )}$ is a topological space, epi-convergence of the sequence ${\displaystyle \{f_{n}\}_{n=1}^{\infty }}$ 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 ${\displaystyle X}$, 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

• Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group. pp. xii+340.

• Kuratowski, Kazimierz (1966). Topology. Volumes I and II. New edition, revised and augmented. Translated from the French by J. Jaworowski. New York: Academic Press. pp. xx+560. MR0217751
• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (1998). Variational analysis. Berlin. ISBN 978-3-642-02431-3. OCLC 883392544.{{cite book}}: CS1 maint: location missing publisher (link)