Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (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.

Isometric, proper, and cobounded actions on metric spaces

Definition

Let G act on a metric space (X,d) by a left action (Left group actions, transitive actions, and faithful actions, Metric space: d(x,y)=0 iff x=y, symmetry, and the triangle inequality; pseudometric and ultrametric).

The action is isometric if every g∈G acts by an isometry, that is, d(g⋅x, g⋅y)=d(x,y)for all x,y∈X.

The action is proper if for every bounded subsets B,C⊆X, the transporter set { g∈G:(g⋅B)∩C≠∅ } is finite.

The action is cobounded if some bounded subset B⊆X has G⋅B:=⋃g∈Gg⋅B=X. If the action is isometric and X is nonempty, coboundedness is equivalent to the existence of x0∈X and R≥0 such that every point of X lies within distance at most R of the orbit G⋅x0.

Depends on

Used by

Dependency tree · two levels

9 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