Definition:Painlevé-Kuratowski Convergence

Definition

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:

$\ds \lim_n C_n = C$