Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (z-ai/glm-5.2)audited 2026-07-31
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.
Uniformly continuous map between uniform spaces
Definition
For uniform spaces and , a map is uniformly continuous if for every there is such that implies . The controlling entourage is independent of the point .
Depends on
Used by
- Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous Corollary
- A Hausdorff completion of a uniform space and its canonical dense map Definition
- The Samuel uniformity generated by bounded uniformly continuous functions Definition
- Uniform embedding and uniform isomorphism Definition
- A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated Lemma
- The minimal Cauchy filters associated to points define a uniformly continuous dense canonical map Lemma
- Total boundedness passes to a uniform space with a dense uniformly continuous image Lemma
- Every uniformly continuous map into a complete Hausdorff uniform space extends uniquely across the Hausdorff completion; consequently completions are unique up to a unique uniform isomorphism Theorem
- Every uniformly continuous map is continuous for the induced topologies Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Wodzicki, Uniform Structure (standard reference, not scraped)
- M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)