TheoremStatement: AI-adaptedProof: AI-adaptedprecheck 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 $G$ .

[L1]

The left and right balls are $L_{U} [x] = x U$ and $R_{U} [x] = U x$ (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 $U$ ranges over neighbourhoods of $e$ , $x U$ ranges over neighbourhoods of $x$ by the left translation homeomorphism, so left balls induce the given topology.

L1L2L3

1.2

Similarly $U x$ ranges over neighbourhoods of $x$ by right translation, so right balls induce the given topology.

L1L2L3

1.3

The identity $(x^{- 1})^{- 1} y^{- 1} = (y x^{- 1})^{- 1}$ 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

Every topological group is uniformizable, and assuming dependent choice it is completely regular Corollary

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)