# Definition:Outer Limit

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## 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 **outer limit** of $\left \langle {C_n}\right \rangle_{n \in \N}$ is defined as:

- $\displaystyle \limsup_{n \to\infty} \ C_n := \left\{{x : \exists N \text{ cofinal set of }\N, \exists x_v \in C_v \left({v \in N}\right) \text{ such that } x_v \to x}\right\}$

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.