DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Uniform space in the entourage formulation

Let $X$ be a set and write $Δ_{X} = {(x, x) : x \in X}$ for its diagonal (The diagonal $Δ_{X} \subseteq X \times X$ , the diagonal map $δ_{X}$ , and the pairing $⟨ f, g ⟩$ of two maps). For $E \subseteq X \times X$ , put $E [x] : = {y \in X : (x, y) \in E}$ , $E^{- 1} : = {(y, x) : (x, y) \in E}$ , and $E \circ F : = {(x, z) : some y has (x, y) \in E, (y, z) \in F}$ .

A uniformity on $X$ is a filter $U$ on $X \times X$ (Filter on a set) such that:

Its members are entourages. A uniform space is a set equipped with a uniformity. The induced topology and its neighbourhoods are constructed in The sets containing an entourage ball about each of their points form a topology.

Depends on