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.
Separated uniformity: the intersection of all entourages is the diagonal
Definition
A uniformity on is separated when (The diagonal , the diagonal map , and the pairing of two maps). Equivalently, whenever , some entourage satisfies . Separation is a property of the uniformity, not an additional convention in the meaning of uniform space.
Depends on
Used by
- A Hausdorff completion of a uniform space and its canonical dense map Definition
- Uniformizable and separated-uniformizable topological spaces Definition
- The Samuel reflection of a nonempty indiscrete uniform space is a singleton Example
- 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 standard entourages on minimal Cauchy filters form a separated uniformity Lemma
- A uniformity is separated if and only if its induced topology is Hausdorff Theorem
- Every countably based uniformity is generated by one pseudometric, which is a metric exactly when the 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: 24 results over 12 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)