ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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The left, right, upper and Roelcke uniformities of the additive topological group $\mathbb{R}$ all equal its metric uniformity

Example

Under addition, $\mathbb R$ is a topological group. Its left, right, upper, and Roelcke uniformities all equal the uniformity generated by $|x-y|<\varepsilon$ .

Facts & Assumptions

Given: The additive group $\mathbb R$ with its usual topology.

[L1]

The group formulas for the left and right entourages are $x^{-1}y\in U$ and $yx^{-1}\in U$ (The left and right uniformities of a topological group, Group and abelian group).

[L2]

The upper and Roelcke structures are respectively generated by intersections and composites of left and right entourages (The upper and Roelcke uniformities generated from the left and right uniformities of a topological group).

[L3]

The usual metric entourages are $|x-y|<\varepsilon$ (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).

Verification

technique · direct

1.1

In additive notation, both left and right conditions are $y-x\in U$ ; for symmetric interval neighbourhoods $U=(-\varepsilon,\varepsilon)$ this is $|x-y|<\varepsilon$ .

L1L3

2.1

Hence the left and right uniformities equal the metric uniformity.

step 1.1

3.1

Their upper join and Roelcke meet equal that same uniformity because the two inputs already agree.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 46 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

M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)
C. Rosendal, Coarse Geometry of Topological Groups (standard reference, not scraped)

The left, right, upper and Roelcke uniformities of the additive topological group R\mathbb{R}R all equal its metric uniformity