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.
where the sequence eventually becomes arbitrarily close to
where the sequence frequently becomes arbitrarily close to
The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space
{\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}}}
{\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
, the limits inferior and superior follow a similar construction.
the collection of open neighborhoods of
The Kuratowski limit inferior of
and the Kuratowski limit superior is the set
are called limit points of
are called cluster points of
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.
be a set-valued function between the spaces
{\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}}}
means convergence in sequences when
is metrizable and convergence in nets otherwise.
Continuity of set-valued functions is commonly defined in terms of lower- and upper-hemicontinuity popularized by Berge.
is continuous with respect to the Vietoris hyperspace topology of
For set-valued functions with closed values, continuity in the sense of Vietoris-Berge is stronger than continuity in the sense of Kuratowski.
defined by the epigraph equation
defined by the epigraph equation
Since Kuratowski upper and lower limits are closed sets, it follows that both
are lower semi-continuous functions.
exists, and the associated function is called the epi-limit of
is a topological space, epi-convergence of the sequence
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
, which does not hold in topological spaces generally.