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

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FALSE: every uniformizable topology has a unique compatible uniformity

Statement

FALSE. Every uniformizable topology has a unique compatible uniformity.

Facts & Assumptions

Given: The usual topology and usual metric uniformity $\mathcal U$ on $\mathbb R$ .

[L1]

The map $h(x)=x/(1+|x|)$ is a homeomorphism $\mathbb R\to(-1,1)$ but not a uniform isomorphism for the usual metric uniformities (The map $x\mapsto x/(1+|x|)$ is a uniformly continuous homeomorphism from $\mathbb{R}$ to $(-1,1)$ whose inverse is not uniformly continuous).

[L2]

Uniformizable means induced by a uniformity, and a uniform isomorphism has uniformly continuous inverse (Uniformizable and separated-uniformizable topological spaces, Uniform embedding and uniform isomorphism).

Refutation

technique · direct

1.1

Pull the usual uniformity of $(-1,1)$ back along the homeomorphism $h$ , obtaining a uniformity $\mathcal V$ on the underlying set $\mathbb R$ .

2.1

Since $h$ is a homeomorphism, $\mathcal V$ induces the usual topology of $\mathbb R$ , so that topology is uniformizable.

step 1.1L2

2.2

If $\mathcal V=\mathcal U$ , then $h$ would be a uniform isomorphism from $\mathcal U$ onto the usual uniformity of $(-1,1)$ , contrary to [L1].

step 1.1L1L2

3.1

Thus one topology has distinct compatible uniformities, refuting the statement.

step 2.1step 2.2∎

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: 37 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)