DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Separated nets in a metric space

Definition

Let $(X, d)$ be a metric space and let $A \subseteq X$ .

For a real $δ > 0$ , the subset $A$ is $δ$ -separated if

$d (a, a^{'}) \geq δ for all distinct a, a^{'} \in A .$

For a real $R \geq 0$ , the subset $A$ is an $R$ -net in $X$ if for every $x \in X$ there is an $a \in A$ with $d (x, a) \leq R$ . For nonempty $A$ this is equivalent, after enlarging $R$ if needed, to a uniform bound on the point-to-set distance of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space. The empty subset is therefore an $R$ -net exactly when $X$ is empty.

A separated net in $X$ is a subset that is $δ$ -separated for some $δ > 0$ and is an $R$ -net for some $R \geq 0$ .

Depends on

Used by

Two metric spaces are quasi-isometric if and only if each contains a separated net and the two nets are bilipschitz equivalent Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Drutu and M. Kapovich, Geometric Group Theory, Section 8.1 (standard reference, not scraped)