ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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The map $x\mapsto x/(1+|x|)$ is a uniformly continuous homeomorphism from $\mathbb{R}$ to $(-1,1)$ whose inverse is not uniformly continuous

Example

The function $h(x)=x/(1+|x|)$ maps $\mathbb R$ onto $(-1,1)$ with inverse $h^{-1}(t)=t/(1-|t|)$ . It is uniformly continuous, but its inverse is not.

Facts & Assumptions

Given: The usual metric uniformities on $\mathbb R$ and $(-1,1)$ .

[L1]

The metric dictionary translates metric uniform continuity into 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).

[L2]

Absolute value is nonnegative (Basic properties of the absolute value) and satisfies the triangle inequality (The triangle inequality).

[L3]

The reciprocal form of the Archimedean property says that $1/n\to0$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ).

Verification

technique · direct

1.1

Direct algebra gives $|h(x)-h(y)|\le2|x-y|$ , so $h$ is uniformly continuous; its displayed inverse and the usual open-interval formulas make it a homeomorphism.

L1L2

1.2

Put $a_n=n/(n+1)$ and $b_n=(n+1)/(n+2)$ . Then $|a_n-b_n|\to0$ , while $|h^{-1}(a_n)-h^{-1}(b_n)|=1$ .

L2L3

2.1

Thus $h^{-1}$ is not uniformly continuous, so this homeomorphism is not a uniform isomorphism (Uniform embedding and uniform isomorphism, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

step 1.1step 1.2∎

Depends on

Used by

FALSE: every uniformizable topology has a unique compatible uniformity False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 56 results over 13 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.

The map x↦x/(1+∣x∣)x\mapsto x/(1+|x|)x↦x/(1+∣x∣) is a uniformly continuous homeomorphism from R\mathbb{R}R to (−1,1)(-1,1)(−1,1) whose inverse is not uniformly continuous