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.
Complete uniform space: every Cauchy filter converges
Definition
A uniform space is complete when every Cauchy filter (Cauchy filter in a uniform space) converges to at least one point of its induced topology (Convergence and cluster points of a filter on a topological space). No separatedness is built into this definition.
Depends on
Used by
- A Hausdorff completion of a uniform space and its canonical dense map Definition
- Every compact uniform space is complete Lemma
- The uniform space of minimal Cauchy filters is complete Lemma
- Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact 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: 11 results over 5 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, Complete uniform space (standard reference, not scraped)