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.