\begin{definition}[Definition:Lower Limit (Topological Space)]
Let $\struct {S, \tau}$ be a topological space.
Let $f: S \to \R \cup \set {-\infty, \infty}$ be an extended real-valued function.
The '''lower limit''' of $f$ at some $x_0 \in S$ is defined as:
:$\ds \liminf_{x \mathop \to x_0} \map f x := \sup_{V \mathop \in \map \mho {x_0} } \set {\inf_{x \mathop \in V} \map f x}$
where $\map \mho {x_0}$ stands for the set of open neighborhoods of $x_0$.
\end{definition}