Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31
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The left and right uniformities of a topological group induce its topology, and inversion interchanges them

Statement

The left and right uniformities of a topological group induce its given topology. Inversion is a uniform isomorphism from the left uniformity to the right uniformity.

Facts & Assumptions

Given: A topological group GG.

[L1]

The left and right balls are LU[x]=xUL_U[x]=xU and RU[x]=UxR_U[x]=Ux (The left and right uniformities of a topological group).

[L2]

Translations and inversion are homeomorphisms (Left and right translations and inversion in a topological group are homeomorphisms).

[L3]

Entourage balls form bases for the induced topologies (The sets containing an entourage ball about each of their points form a topology).

Proof

technique · direct
1.1

As UU ranges over neighbourhoods of ee, xUxU ranges over neighbourhoods of xx by the left translation homeomorphism, so left balls induce the given topology.

L1L2L3
1.2

Similarly UxUx ranges over neighbourhoods of xx by right translation, so right balls induce the given topology.

L1L2L3
1.3

The identity (x−1)−1y−1=(yx−1)−1(x^{-1})^{-1}y^{-1}=(yx^{-1})^{-1} sends a left entourage to the inverse of the corresponding right entourage; shrinking neighbourhoods proves uniform continuity in both directions.

L1L2
2.1

Thus inversion interchanges the two uniformities as a uniform isomorphism.

step 1.3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 11 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