\begin{definition}[Definition:Painlevé-Kuratowski Convergence]
Let $T = \struct {S, \tau}$ be a Hausdorff topological space.
Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $T$.
Let $\sequence {C_n}_{n \mathop \in \N}$ be such that:
:$\ds \liminf_n C_n = \limsup_n C_n = C$
where:
:$\ds \liminf_n C_n$ denotes the inner limit of $\sequence {C_n}_{n \mathop \in \N}$
:$\ds \limsup_n C_n$ denotes the outer limit of $\sequence {C_n}_{n \mathop \in \N}$
Then $\sequence {C_n}_{n \mathop \in \N}$ 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:
:$\ds \lim_n C_n = C$
{{NamedforDef|Paul Painlevé|name2 = Kazimierz Kuratowski|cat = Painlevé|cat2 = Kuratowski}}
Category:Definitions/Hausdorff Spaces
Category:Definitions/Convergence
\end{definition}