Definition:Lower Limit (Topological Space)

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This page is about the lower limit of a point in a topological space. For other uses, see Definition:Lower Limit.

Definition

Let $\left({S, \tau}\right)$ be a topological space.

Let $f: S \to \R \cup \left\{{-\infty, \infty}\right\}$ be an extended real-valued function.


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

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

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


Note


Because $x_0 \in V$ for all $V \mathop \in \mho \left({x_0}\right)$, the above definition implies:

$\displaystyle \liminf_{x \mathop \to x_0}\ f \left({x}\right) \le f \left({x_0}\right)$


Also defined as

Some authors exclude $x_0$ from the infimum in the definition:

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

These definitions differ only if $f \left({x}\right)$ is discontinuous at $x_0$.


Also see