Definition:Painlevé-Kuratowski Convergence

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:
 * $\displaystyle \lim_n C_n = C$