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 edit

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

Topological Spaces edit

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 neighbhorhoods of ${\textstyle x}$ . The Kuratowski limit inferior of ${\textstyle \{A_{i}\}_{i\in I}}$  is the set

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

• 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 ${\displaystyle \{(0,y)\in \mathbb {R} ^{2}:y\geq 0\}}$ .

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.

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

• ${\displaystyle S}$  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 edit

• 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 edit

• 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.

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

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.

• 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

• 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)