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$${\displaystyle {\begin{aligned}\mathop {\mathrm {Li} } A_{n}:=&\left\{x\in X:{\begin{matrix}{\mbox{for all open neighbourhoods }}U{\mbox{ of }}x,U\cap A_{n}\neq \emptyset {\mbox{ for large enough }}n\end{matrix}}\right\}\\=&\left\{x\in X:\limsup _{n\to \infty }d(x,A_{n})=0\right\};\end{aligned}}}$$the Kuratowski limit superior (or upper closed limit) of ${\displaystyle A_{n}}$ as ${\displaystyle n\to \infty }$; is$${\displaystyle {\begin{aligned}\mathop {\mathrm {Ls} } A_{n}:=&\left\{x\in X:{\begin{matrix}{\mbox{for all open neighbourhoods }}U{\mbox{ of }}x,U\cap A_{n}\neq \emptyset {\mbox{ for infinitely many }}n\end{matrix}}\right\}\\=&\left\{x\in X:\liminf _{n\to \infty }d(x,A_{n})=0\right\};\end{aligned}}}$$If 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$${\displaystyle \mathop {\mathrm {Li} } A_{i}:=\left\{x\in X:{\mbox{for all }}U\in {\mathcal {N}}(x){\mbox{ there exists }}i_{0}\in I{\mbox{ such that }}U\cap A_{i}\neq \emptyset {\text{ if }}i_{0}\leq i\right\},}$$and the Kuratowski limit superior is the set$${\displaystyle \mathop {\mathrm {Ls} } A_{i}:=\left\{x\in X:{\mbox{for all }}U\in {\mathcal {N}}(x){\mbox{ and }}i\in I{\mbox{ there exists }}i'\in I{\mbox{ such that }}i\leq i'{\mbox{ and }}U\cap A_{i'}\neq \emptyset \right\}.}$$Elements 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 ${\displaystyle A\cap K=\emptyset }$ there exists ${\displaystyle n_{0}}$ such that ${\displaystyle A_{n}\cap K=\emptyset }$ 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$${\displaystyle {\begin{aligned}\mathop {\mathrm {Li} } _{x'\to x}S(x'):=&\bigcap _{x'\to x}\mathop {\mathrm {Li} } S(x'),\qquad x\in X\\\mathop {\mathrm {Ls} } _{x'\to x}S(x'):=&\bigcup _{x'\to x}\mathop {\mathrm {Ls} } S(x'),\qquad x\in X\\\end{aligned}}}$$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

Main articles: 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$${\displaystyle \mathop {\mathrm {epi} } \left(\mathop {\mathrm {e} \liminf } f_{n}\right):=\mathop {\mathrm {Ls} } \left(\mathop {\mathrm {epi} } f_{n}\right),}$$and similarly the epi-limit superior (or upper epi-limit) is the function ${\displaystyle \mathop {\mathrm {e} \limsup } f_{n}}$ defined by the epigraph equation$${\displaystyle \mathop {\mathrm {epi} } \left(\mathop {\mathrm {e} \limsup } f_{n}\right):=\mathop {\mathrm {Li} } \left(\mathop {\mathrm {epi} } f_{n}\right).}$$Since 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

1. This is reported in the Commentary section of Chapter 4 of Rockafellar and Wets' text.
2. Hausdorff, Felix (1927). Mengenlehre (in German) (2nd ed.). Berlin: Walter de Gruyter & Co.
3. Kuratowski, Kazimierz (1933). Topologie, I & II (in French). Warsaw: Panstowowe Wyd Nauk.
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 forced 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 ${\displaystyle S}$ with unbounded range and even unbounded value sets ${\displaystyle S(x)}$ 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

• 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. MR 0217751
• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (1998). Variational analysis. Berlin. ISBN   978-3-642-02431-3. OCLC   883392544.