Definition:Net (Metric Space)

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Definition

Let $M = \left({A, d}\right)$ 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:

$\displaystyle A \subseteq \bigcup_{x \mathop \in S} B_\epsilon \left({x}\right)$

where $B_\epsilon \left({x}\right)$ 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.


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