Definition:Net (Metric Space)

From ProofWiki
Jump to navigation Jump to search

Definition

Let $M = \struct {A, d}$ be a metric space.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.


An $\epsilon$-net for $M$ is a subset $S \subseteq A$ such that:

$\displaystyle A \subseteq \bigcup_{x \mathop \in S} \map {B_\epsilon} x$

where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.


That is, it is a subset of $A$ such that the set of all open $\epsilon$-balls of elements of that set forms a cover for $A$.


Finite Net

A finite $\epsilon$-net for $M$ is an $\epsilon$-net for $M$ which is a finite set.


Sources