DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Extended diameter for Hausdorff covers

Definition

For a metric space $(X, d)$ extend the diameter of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space to every subset $A \subseteq X$ by

$diam A = {\begin{cases} 0 & A = \emptyset, \\ \sup {d (x, y) : x, y \in A} & A \neq \emptyset and A bounded, \\ + \infty & A unbounded . \end{cases}$

The bounded nonempty branch is the existing real diameter; the infinite value belongs to The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined. A nonempty set has diameter zero exactly when it is a singleton: all its pairwise distances must be zero. No value of a point-to-empty-set distance is introduced.

Depends on

Used by

Hausdorff content at a prescribed scale Definition

Dependency tree · two levels

17 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

Fremlin 264A, 264K (standard reference, not scraped)