Definition:Net (Metric Space)

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:

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

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

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Definition $7.2.8$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces