Definition:Net (Metric Space)
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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