Alphabeta MathHow 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.
Uniform embedding and uniform isomorphism
Definition
A map of uniform spaces is a uniform embedding if it is injective and its corestriction , with the subspace uniformity, is a uniform isomorphism. A uniform isomorphism is a bijection whose map and inverse are uniformly continuous. Bijection and corestriction are understood in the sense of Injection, surjection, bijection.
Depends on
Used by
- The Samuel compactification map need not be a uniform embedding for the original uniformity Counterexample
- A Hausdorff completion of a uniform space and its canonical dense map Definition
- The map x↦ x/(1+|x|) is a uniformly continuous homeomorphism from ℝ to (-1,1) whose inverse is not uniformly continuous Example
- FALSE: every uniformizable topology has a unique compatible uniformity False statement
- Every uniform space has a Hausdorff completion with dense canonical image, and the canonical map is a uniform embedding exactly when the original uniformity is separated Theorem
- 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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 14 results over 6 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)
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)