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, the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".
Definitions For a given sequence { x n } n = 1 ∞ {\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 ∈ X {\displaystyle x\in 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 ∈ X {\displaystyle x\in 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 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 ⊂ X {\displaystyle A\subset X} , define the distance between the point and the subset:
d ( x , A ) := inf y ∈ A d ( x , y ) , x ∈ X . {\displaystyle d(x,A):=\inf _{y\in A}d(x,y),\qquad x\in X.}
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