[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki4\/kuratowski-convergence-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki4\/kuratowski-convergence-wikipedia\/","headline":"Kuratowski convergence – Wikipedia","name":"Kuratowski convergence – Wikipedia","description":"In mathematics, Kuratowski convergence or Painlev\u00e9-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced","datePublished":"2016-03-23","dateModified":"2016-03-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki4\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki4\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/02d3ffd73dbe0cdd90a51b461341f72fdc95734d","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/02d3ffd73dbe0cdd90a51b461341f72fdc95734d","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki4\/kuratowski-convergence-wikipedia\/","about":["Wiki"],"wordCount":21439,"articleBody":"In mathematics, Kuratowski convergence or Painlev\u00e9-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlev\u00e9 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”.Table of ContentsDefinitions[edit]Metric Spaces[edit]Topological Spaces[edit]Examples[edit]Properties[edit]Kuratowski Continuity of Set-Valued Functions[edit]Examples[edit]Properties[edit]Epi-convergence and \u0393-convergence[edit]See also[edit]References[edit]Definitions[edit]For a given sequence {xn}n=1\u221e{displaystyle {x_{n}}_{n=1}^{infty }} of points in a space X{displaystyle X}, a limit point of the sequence can be understood as any point x\u2208X{displaystyle xin X} where the sequence eventually becomes arbitrarily close to x{displaystyle x}. On the other hand, a cluster point of the sequence can be thought of as a point x\u2208X{displaystyle xin X} where the sequence frequently becomes arbitrarily close to x{displaystyle x}. The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space X{displaystyle X}.Metric Spaces[edit]Let (X,d){displaystyle (X,d)} be a metric space, where X{displaystyle X} is a given set. For any point x{displaystyle x} and any non-empty subset A\u2282X{displaystyle Asubset X}, define the distance between the point and the subset:d(x,A):=infy\u2208Ad(x,y),x\u2208X.{displaystyle d(x,A):=inf _{yin A}d(x,y),qquad xin X.}For any sequence of subsets {An}n=1\u221e{displaystyle {A_{n}}_{n=1}^{infty }} of X{displaystyle X}, the Kuratowski limit inferior (or lower closed limit) of An{displaystyle A_{n}} as n\u2192\u221e{displaystyle nto infty }; isLi\u2061An:={x\u2208X:for all open neighbourhoods\u00a0U\u00a0of\u00a0x,U\u2229An\u2260\u2205\u00a0for large enough\u00a0n}={x\u2208X:lim\u2006supn\u2192\u221ed(x,An)=0};{displaystyle {begin{aligned}mathop {mathrm {Li} } A_{n}:=&left{xin X:{begin{matrix}{mbox{for all open neighbourhoods }}U{mbox{ of }}x,Ucap A_{n}neq emptyset {mbox{ for large enough }}nend{matrix}}right}\\=&left{xin X:limsup _{nto infty }d(x,A_{n})=0right};end{aligned}}}the Kuratowski limit superior (or upper closed limit) of An{displaystyle A_{n}} as n\u2192\u221e{displaystyle nto infty }; isLs\u2061An:={x\u2208X:for all open neighbourhoods\u00a0U\u00a0of\u00a0x,U\u2229An\u2260\u2205\u00a0for infinitely many\u00a0n}={x\u2208X:lim\u2006infn\u2192\u221ed(x,An)=0};{displaystyle {begin{aligned}mathop {mathrm {Ls} } A_{n}:=&left{xin X:{begin{matrix}{mbox{for all open neighbourhoods }}U{mbox{ of }}x,Ucap A_{n}neq emptyset {mbox{ for infinitely many }}nend{matrix}}right}\\=&left{xin X:liminf _{nto infty }d(x,A_{n})=0right};end{aligned}}}If the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of An{displaystyle A_{n}} and is denoted Limn\u2192\u221e\u2061An{displaystyle mathop {mathrm {Lim} } _{nto infty }A_{n}}.Topological Spaces[edit]If (X,\u03c4){textstyle (X,tau )} is a topological space, and {Ai}i\u2208I{textstyle {A_{i}}_{iin I}} are a net of subsets of X{textstyle X}, the limits inferior and superior follow a similar construction. For a given point x\u2208X{textstyle xin X} denote N(x){textstyle {mathcal {N}}(x)} the collection of open neighbhorhoods of x{textstyle x}. The Kuratowski limit inferior of {Ai}i\u2208I{textstyle {A_{i}}_{iin I}} is the setLi\u2061Ai:={x\u2208X:for all\u00a0U\u2208N(x)\u00a0there exists\u00a0i0\u2208I\u00a0such that\u00a0U\u2229Ai\u2260\u2205\u00a0if\u00a0i0\u2264i},{displaystyle mathop {mathrm {Li} } A_{i}:=left{xin X:{mbox{for all }}Uin {mathcal {N}}(x){mbox{ there exists }}i_{0}in I{mbox{ such that }}Ucap A_{i}neq emptyset {text{ if }}i_{0}leq iright},}and the Kuratowski limit superior is the setLs\u2061Ai:={x\u2208X:for all\u00a0U\u2208N(x)\u00a0and\u00a0i\u2208I\u00a0there exists\u00a0i\u2032\u2208I\u00a0such that\u00a0i\u2264i\u2032\u00a0and\u00a0U\u2229Ai\u2032\u2260\u2205}.{displaystyle mathop {mathrm {Ls} } A_{i}:=left{xin X:{mbox{for all }}Uin {mathcal {N}}(x){mbox{ and }}iin I{mbox{ there exists }}i’in I{mbox{ such that }}ileq i'{mbox{ and }}Ucap A_{i’}neq emptyset right}.}Elements of Li\u2061Ai{textstyle mathop {mathrm {Li} } A_{i}} are called limit points of {Ai}i\u2208I{textstyle {A_{i}}_{iin I}} and elements of Ls\u2061Ai{textstyle mathop {mathrm {Ls} } A_{i}} are called cluster points of {Ai}i\u2208I{textstyle {A_{i}}_{iin I}}. In other words, x{displaystyle x} is a limit point of {Ai}i\u2208I{textstyle {A_{i}}_{iin I}} if each of its neighborhoods intersects Ai{displaystyle A_{i}} for all i{displaystyle i} in a “residual” subset of I{displaystyle I}, while x{displaystyle x} is a cluster point of {Ai}i\u2208I{textstyle {A_{i}}_{iin I}} if each of its neighborhoods intersects Ai{displaystyle A_{i}} for all i{displaystyle i} in a cofinal subset of I{displaystyle I}.When these sets agree, the common set is the Kuratowski limit of {Ai}i\u2208I{textstyle {A_{i}}_{iin I}}, denoted Lim\u2061Ai{displaystyle mathop {mathrm {Lim} } A_{i}}.Examples[edit]Suppose (X,d){displaystyle (X,d)} is separable where X{displaystyle X} is a perfect set, and let D={d1,d2,\u2026}{displaystyle D={d_{1},d_{2},dots }} be an enumeration of a countable dense subset of X{displaystyle X}. Then the sequence {An}n=1\u221e{displaystyle {A_{n}}_{n=1}^{infty }} defined by An:={d1,d2,\u2026,dn}{displaystyle A_{n}:={d_{1},d_{2},dots ,d_{n}}} has Lim\u2061An=X{displaystyle mathop {mathrm {Lim} } A_{n}=X}.Given two closed subsets B,C\u2282X{displaystyle B,Csubset X}, defining A2n\u22121:=B{displaystyle A_{2n-1}:=B} and A2n:=C{displaystyle A_{2n}:=C} for each n=1,2,\u2026{displaystyle n=1,2,dots } yields Li\u2061An=B\u2229C{displaystyle mathop {mathrm {Li} } A_{n}=Bcap C} and Ls\u2061An=B\u222aC{displaystyle mathop {mathrm {Ls} } A_{n}=Bcup C}.The sequence of closed balls An:={y\u2208X:d(xn,y)\u2264rn}{displaystyle A_{n}:={yin X:d(x_{n},y)leq r_{n}}}converges in the sense of Kuratowski when xn\u2192x{displaystyle x_{n}to x} in X{displaystyle X} and rn\u2192r{displaystyle r_{n}to r} in [0,+\u221e){displaystyle [0,+infty )}, and in particular, Lim\u2061(An)={y\u2208X:d(x,y)\u2264r}{displaystyle mathop {mathrm {Lim} } (A_{n})={yin X:d(x,y)leq r}}. If rn\u2192+\u221e{displaystyle r_{n}to +infty }, then Lim\u2061An=X{displaystyle mathop {mathrm {Lim} } A_{n}=X} while Lim\u2061(X\u2216An)=\u2205{displaystyle mathop {mathrm {Lim} } (Xsetminus A_{n})=emptyset }.Let An:={x\u2208R:sin\u2061(nx)=0}{textstyle A_{n}:={xin mathbb {R} :sin(nx)=0}}. Then An{displaystyle A_{n}} converges in the Kuratowski sense to the entire line.In a topological vector space, if {An}n=1\u221e{displaystyle {A_{n}}_{n=1}^{infty }} is a sequence of cones, then so are the Kuratowski limits superior and inferior. For example, the sets An:={(x,y)\u2208R2:y\u2265n|x|}{displaystyle A_{n}:={(x,y)in mathbb {R} ^{2}:ygeq n|x|}} converge to {(0,y)\u2208R2:y\u22650}{displaystyle {(0,y)in mathbb {R} ^{2}:ygeq 0}}.Properties[edit]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\u2061An{displaystyle mathop {mathrm {Li} } A_{n}} and Ls\u2061An{displaystyle mathop {mathrm {Ls} } A_{n}} are closed subsets of X{displaystyle X}, and Li\u2061An\u2282Ls\u2061An{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: Li\u2061An=Li\u2061cl\u2061(An){displaystyle mathop {mathrm {Li} } A_{n}=mathop {mathrm {Li} } mathop {mathrm {cl} } (A_{n})} and Ls\u2061An=Ls\u2061cl\u2061(An){displaystyle mathop {mathrm {Ls} } A_{n}=mathop {mathrm {Ls} } mathop {mathrm {cl} } (A_{n})}.If An:=A{displaystyle A_{n}:=A} is a constant sequence, then Lim\u2061An=cl\u2061A{displaystyle mathop {mathrm {Lim} } A_{n}=mathop {mathrm {cl} } A}.If An:={xn}{displaystyle A_{n}:={x_{n}}} is a sequence of singletons, then Li\u2061An{displaystyle mathop {mathrm {Li} } A_{n}} and Ls\u2061An{displaystyle mathop {mathrm {Ls} } A_{n}} consist of the limit points and cluster points, respectively, of the sequence {xn}n=1\u221e\u2282X{displaystyle {x_{n}}_{n=1}^{infty }subset X}.If An\u2282Bn\u2282Cn{displaystyle A_{n}subset B_{n}subset C_{n}} and B:=Lim\u2061An=Lim\u2061Cn{displaystyle B:=mathop {mathrm {Lim} } A_{n}=mathop {mathrm {Lim} } C_{n}}, then Lim\u2061Bn=B{displaystyle mathop {mathrm {Lim} } B_{n}=B}.(Hit and miss criteria) For a closed subset A\u2282X{displaystyle Asubset X}, one hasIf A1\u2282A2\u2282A3\u2282\u22ef{displaystyle A_{1}subset A_{2}subset A_{3}subset cdots } then the Kuratowski limit exists, and Lim\u2061An=cl\u2061(\u22c3n=1\u221eAn){textstyle mathop {mathrm {Lim} } A_{n}=mathop {mathrm {cl} } left(bigcup _{n=1}^{infty }A_{n}right)}. Conversely, if A1\u2283A2\u2283A3\u2283\u22ef{displaystyle A_{1}supset A_{2}supset A_{3}supset cdots } then the Kuratowski limit exists, and Lim\u2061An=\u22c2n=1\u221ecl\u2061(An){textstyle mathop {mathrm {Lim} } A_{n}=bigcap _{n=1}^{infty }mathop {mathrm {cl} } (A_{n})}.If dH{displaystyle d_{H}} denotes Hausdorff metric, then dH(An,A)\u21920{displaystyle d_{H}(A_{n},A)to 0} implies cl\u2061A=Lim\u2061An{displaystyle mathop {mathrm {cl} } A=mathop {mathrm {Lim} } A_{n}}. However, noncompact closed sets may converge in the sense of Kuratowski while dH(An,Lim\u2061An)=+\u221e{displaystyle d_{H}(A_{n},mathop {mathrm {Lim} } A_{n})=+infty } for each n=1,2,\u2026{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 X{displaystyle X} is compact, then these are all equivalent and agree with convergence in Hausdorff metric.Kuratowski Continuity of Set-Valued Functions[edit]Let S:X\u21c9Y{displaystyle S:Xrightrightarrows Y} be a set-valued function between the spaces X{displaystyle X} and Y{displaystyle Y}; namely, S(x)\u2282Y{displaystyle S(x)subset Y} for all x\u2208X{displaystyle xin X}. Denote S\u22121(y)={x\u2208X:y\u2208S(x)}{displaystyle S^{-1}(y)={xin X:yin S(x)}}. We can define the operatorsLix\u2032\u2192x\u2061S(x\u2032):=\u22c2x\u2032\u2192xLi\u2061S(x\u2032),x\u2208XLsx\u2032\u2192x\u2061S(x\u2032):=\u22c3x\u2032\u2192xLs\u2061S(x\u2032),x\u2208X{displaystyle {begin{aligned}mathop {mathrm {Li} } _{x’to x}S(x’):=&bigcap _{x’to x}mathop {mathrm {Li} } S(x’),qquad xin X\\mathop {mathrm {Ls} } _{x’to x}S(x’):=&bigcup _{x’to x}mathop {mathrm {Ls} } S(x’),qquad xin X\\end{aligned}}}where x\u2032\u2192x{displaystyle x’to x} means convergence in sequences when X{displaystyle X} is metrizable and convergence in nets otherwise. Then,When S{displaystyle S} is both inner and outer semi-continuous at x\u2208X{displaystyle xin X}, we say that S{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 fS:X\u21922Y{displaystyle f_{S}:Xto 2^{Y}} defined by f(x)=S(x){displaystyle f(x)=S(x)} is continuous with respect to the Vietoris hyperspace topology of 2Y{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 B(x,r)={y\u2208X:d(x,y)\u2264r}{displaystyle B(x,r)={yin X:d(x,y)leq r}} is continuous X\u00d7[0,+\u221e)\u21c9X{displaystyle Xtimes [0,+infty )rightrightarrows X}.Given a function f:X\u2192[\u2212\u221e,+\u221e]{displaystyle f:Xto [-infty ,+infty ]}, the superlevel set mapping Sf(x):={\u03bb\u2208R:f(x)\u2264\u03bb}{displaystyle S_{f}(x):={lambda in mathbb {R} :f(x)leq lambda }} is outer semi-continuous at x{displaystyle x}, if and only if f{displaystyle f} is lower semi-continuous at x{displaystyle x}. Similarly, Sf{displaystyle S_{f}} is inner semi-continuous at x{displaystyle x}, if and only if f{displaystyle f} is upper semi-continuous at x{displaystyle x}.Properties[edit]Epi-convergence and \u0393-convergence[edit]For the metric space (X,d){displaystyle (X,d)} a sequence of functions fn:X\u2192[\u2212\u221e,+\u221e]{displaystyle f_{n}:Xto [-infty ,+infty ]}, the epi-limit inferior (or lower epi-limit) is the function elim\u2006inf\u2061fn{displaystyle mathop {mathrm {e} liminf } f_{n}} defined by the epigraph equationepi\u2061(elim\u2006inf\u2061fn):=Ls\u2061(epi\u2061fn),{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 elim\u2006sup\u2061fn{displaystyle mathop {mathrm {e} limsup } f_{n}} defined by the epigraph equationepi\u2061(elim\u2006sup\u2061fn):=Li\u2061(epi\u2061fn).{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 elim\u2006inf\u2061fn{displaystyle mathop {mathrm {e} liminf } f_{n}} and elim\u2006sup\u2061fn{displaystyle mathop {mathrm {e} limsup } f_{n}} are lower semi-continuous functions. Similarly, since Li\u2061epi\u2061fn\u2282Ls\u2061epi\u2061fn{displaystyle mathop {mathrm {Li} } mathop {mathrm {epi} } f_{n}subset mathop {mathrm {Ls} } mathop {mathrm {epi} } f_{n}}, it follows that elim\u2006inf\u2061fn\u2264elim\u2006inf\u2061fn{displaystyle mathop {mathrm {e} liminf } f_{n}leq mathop {mathrm {e} liminf } f_{n}} uniformly. These functions agree, if and only if Lim\u2061epi\u2061fn{displaystyle mathop {mathrm {Lim} } mathop {mathrm {epi} } f_{n}} exists, and the associated function is called the epi-limit of {fn}n=1\u221e{displaystyle {f_{n}}_{n=1}^{infty }}.When (X,\u03c4){displaystyle (X,tau )} is a topological space, epi-convergence of the sequence {fn}n=1\u221e{displaystyle {f_{n}}_{n=1}^{infty }} is called \u0393-convergence. From the perspective of Kuratowski convergence there is no distinction between epi-limits and \u0393-limits. The concepts are usually studied separately, because epi-convergence admits special characterizations that rely on the metric space structure of X{displaystyle X}, which does not hold in topological spaces generally.See also[edit]^ This is reported in the Commentary section of Chapter 4 of Rockafellar and Wets’ text. ^ Hausdorff, Felix (1927). Mengenlehre (in German) (2nd\u00a0ed.). Berlin: Walter de Gruyter & Co.^ Kuratowski, Kazimierz (1933). Topologie, I & II (in French). Warsaw: Panstowowe Wyd Nauk.^ 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.^ For an example, consider the sequence of cones in the previous section.^ 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{displaystyle S} with unbounded range and even unbounded value sets S(x){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[edit]Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group. pp.\u00a0xii+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.\u00a0xx+560. MR0217751"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki4\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki4\/kuratowski-convergence-wikipedia\/#breadcrumbitem","name":"Kuratowski convergence – Wikipedia"}}]}]