DefinitionDefinition: AI-adaptedProof: 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.

The left and right uniformities of a topological group

Definition

Let $G$ be a topological group with identity $e$ . For every neighbourhood $U$ of $e$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open), put

$L_{U} = {(x, y) : x^{- 1} y \in U}, R_{U} = {(x, y) : y x^{- 1} \in U} .$

The filters generated by the $L_{U}$ and by the $R_{U}$ are respectively the left uniformity and the right uniformity of $G$ . These are uniformities. Indeed, finite intersections of identity neighbourhoods refine finite intersections of the corresponding relations; the diagonal lies in every relation; $L_{U}^{- 1} = L_{U^{- 1}}$ and $R_{U}^{- 1} = R_{U^{- 1}}$ ; and continuity of multiplication at $(e, e)$ gives a neighbourhood $V$ with $V \cdot V \subseteq U$ , whence $L_{V} \circ L_{V} \subseteq L_{U}$ and $R_{V} \circ R_{V} \subseteq R_{U}$ . Thus all axioms of Uniform space in the entourage formulation hold.

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)