Definition:Lower Limit (Topological Space)

Definition
Let $\left(S,\tau\right)$ be a topological space and $f:S\to\R\cup\left\{-\infty,\infty\right\}$ be a extended real-valued function.

The lower limit of $f$ at some $x_0 \in S$ is defined as:


 * $\displaystyle \liminf_{x \to x_0}\ f \left({x}\right) := \sup_{V \in \mho \left({x_0}\right)} \left[{\inf_{x \in V} \ f \left({x}\right)}\right]$

where $\mho \left({x_0}\right)$ stands for the family of open neighborhoods of $x_0$.

Disambiguation
The lower limit of a function is a topological property in the sense that it depends on the underlying topology of the space. It should not be confused with the limit inferior.