Definition:Inner Limit

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

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

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