Definition:Painlevé-Kuratowski Convergence

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Definition

Let $T = \left({S, \tau}\right)$ be a Hausdorff topological space.

Let $\left \langle {C_n}\right \rangle_{n \mathop \in \N}$ be a sequence of sets in $T$.


Let $\left \langle {C_n}\right \rangle_{n \mathop \in \N}$ be such that:

$\displaystyle \liminf_n C_n = \limsup_n C_n = C$

where:

$\displaystyle \liminf_n C_n$ denotes the inner limit of $\left \langle {C_n}\right \rangle_{n \mathop \in \N}$
$\displaystyle \limsup_n C_n$ denotes the outer limit of $\left \langle {C_n}\right \rangle_{n \mathop \in \N}$


Then $\left \langle {C_n}\right \rangle_{n \mathop \in \N}$ is said to be convergent in the sense of Painlevé-Kuratowski.


It can be denoted as:

$C_n \overset K \to C$

or:

$\operatorname {K-lim} \limits_{n \mathop \to \infty} C_n = C$

or simply:

$\displaystyle \lim_n C_n = C$


Source of Name

This entry was named for Paul Painlevé and Kazimierz Kuratowski.