Definition:Outer Limit

Definition

Let $\struct {X, \tau}$ be a Hausdorff topological space.

Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $X$.

The outer limit of $\sequence {C_n}_{n \mathop \in \N}$ is defined as:

$\ds \limsup_{n \mathop \to \infty} \ C_n := \set {x \in X: \exists N \text{ cofinal set of }\N, \exists x_v \in C_v \paren {v \in N} \text{ such that } x_v \to x}$

where $x_v \to x$ denotes convergence in the topology $\tau$.

Note

The definition of the inner limit of a sequence of sets extends that of the limit inferior of real numbers to a general topological space.

The outer limit should not be confused with the limit superior of sequence of sets, whose definition assumes no topological structure. Unfortunately, the same symbol $\limsup$ is usually used both for the outer limit as well as for the limit superior, so the distinction needs to be made explicit.