Definition:Painlevé-Kuratowski Convergence

Definition

Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff topological space.

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

The sequence $\left \langle {C_n}\right \rangle_{n \in \N}$ is said to be convergent in the sense of Painlevé-Kuratowski,

denoted as

$C_n \overset{K} \to C$

or

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

or simply

$\lim_n C_n = C$

if $\liminf_n C_n = \limsup_n C_n = C$

where $\liminf_n C_n$ stands for the inner limit of $\left \langle {C_n}\right \rangle_{n \in \N}$ and $\limsup_n C_n$ stands for the outer limit.