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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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.
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The usual metric entourages on $\mathbb{R}$ induce its usual topology and usual uniform continuity

Example

For $\varepsilon>0$ , let $E_\varepsilon=\{(x,y):|x-y|<\varepsilon\}$ on $\mathbb R$ . These are the usual metric entourages.

Facts & Assumptions

Given: The usual metric $d(x,y)=|x-y|$ on $\mathbb R$ .

[L1]

This is a metric and its metric topology is the usual topology (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$ , and it is unbounded).

[L2]

The metric-uniformity dictionary identifies metric and entourage notions of topology and uniform continuity (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, Uniform continuity of a map of metric spaces: one $\delta$ serving every point).

Verification

technique · direct

1.1

The entourage $E_\varepsilon$ is exactly the relation $d(x,y)<\varepsilon$ .

2.1

Applying [L2] gives the usual topology and identifies uniform continuity for these entourages with the usual $\varepsilon$ - $\delta$ condition.

step 1.1L2∎

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: 43 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

J. Wodzicki, Uniform Structure (standard reference, not scraped)
M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)

The usual metric entourages on R\mathbb{R}R induce its usual topology and usual uniform continuity