Kuratowski convergence – Wikipedia

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[edit]

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

${displaystyle xin X}$

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

${displaystyle xin X}$

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

${displaystyle Asubset X}$

, define the distance between the point and the subset:

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

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

${displaystyle nto infty }$

; is

the Kuratowski limit superior (or upper closed limit) of

${displaystyle A_{n}}$

as

${displaystyle nto infty }$

; is

If the Kuratowski limits inferior and superior agree, then the common set is called the Kuratowski limit of

${displaystyle A_{n}}$

and is denoted

${displaystyle mathop {mathrm {Lim} } _{nto infty }A_{n}}$

.

Topological Spaces[edit]

If

${textstyle (X,tau )}$

is a topological space, and

${textstyle {A_{i}}_{iin I}}$

are a net of subsets of

${textstyle X}$

, the limits inferior and superior follow a similar construction. For a given point

${textstyle xin X}$

denote

${textstyle {mathcal {N}}(x)}$

the collection of open neighbhorhoods of

${textstyle x}$

. The Kuratowski limit inferior of

${textstyle {A_{i}}_{iin I}}$

is the set

and the Kuratowski limit superior is the set

Elements of

${textstyle mathop {mathrm {Li} } A_{i}}$

are called limit points of

${textstyle {A_{i}}_{iin I}}$

and elements of

${textstyle mathop {mathrm {Ls} } A_{i}}$

are called cluster points of

${textstyle {A_{i}}_{iin I}}$

. In other words,

${displaystyle x}$

is a limit point of

${textstyle {A_{i}}_{iin I}}$

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

${textstyle {A_{i}}_{iin I}}$

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

${textstyle {A_{i}}_{iin I}}$

, denoted

${displaystyle mathop {mathrm {Lim} } A_{i}}$

.

Examples[edit]

• 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
  ${displaystyle B,Csubset X}$

  , defining

  ${displaystyle A_{2n-1}:=B}$

  and

  ${displaystyle A_{2n}:=C}$

  for each

  ${displaystyle n=1,2,dots }$

  yields

  ${displaystyle mathop {mathrm {Li} } A_{n}=Bcap C}$

  and

  ${displaystyle mathop {mathrm {Ls} } A_{n}=Bcup C}$

  .

• The sequence of closed balls
  ${displaystyle A_{n}:={yin X:d(x_{n},y)leq r_{n}}}$

  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,

  ${displaystyle mathop {mathrm {Lim} } (A_{n})={yin X:d(x,y)leq r}}$

  . If

  ${displaystyle r_{n}to +infty }$

  , then

  ${displaystyle mathop {mathrm {Lim} } A_{n}=X}$

  while

  ${displaystyle mathop {mathrm {Lim} } (Xsetminus A_{n})=emptyset }$

  .

• Let
  ${textstyle A_{n}:={xin mathbb {R} :sin(nx)=0}}$

  . 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

  ${displaystyle A_{n}:={(x,y)in mathbb {R} ^{2}:ygeq n|x|}}$

  converge to

  ${displaystyle {(0,y)in mathbb {R} ^{2}:ygeq 0}}$

  .

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
  ${displaystyle mathop {mathrm {Li} } A_{n}}$

  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
  ${displaystyle Asubset X}$

  , one has

• 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[edit]

Let

${displaystyle S:Xrightrightarrows Y}$

be a set-valued function between the spaces

${displaystyle X}$

and

${displaystyle Y}$

; namely,

${displaystyle S(x)subset Y}$

for all

${displaystyle xin X}$

. Denote

${displaystyle S^{-1}(y)={xin X:yin S(x)}}$

. 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,

When

${displaystyle S}$

is both inner and outer semi-continuous at

${displaystyle xin X}$

, 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

${displaystyle f_{S}:Xto 2^{Y}}$

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
  ${displaystyle B(x,r)={yin X:d(x,y)leq r}}$

  is continuous

  ${displaystyle Xtimes [0,+infty )rightrightarrows X}$

  .

• Given a function
  ${displaystyle f:Xto [-infty ,+infty ]}$

  , 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]

Epi-convergence and Γ-convergence[edit]

For the metric space

${displaystyle (X,d)}$

a sequence of functions

${displaystyle f_{n}:Xto [-infty ,+infty ]}$

, 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 equation

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[edit]

References[edit]

• 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