Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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 U on R.

[L1]

The map h(x)=x/(1+∣x∣) is a homeomorphism R→(−1,1) but not a uniform isomorphism for the usual metric uniformities (The map x↦x/(1+∣x∣) is a uniformly continuous homeomorphism from 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 V on the underlying set R.

L1
2.1

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

step 1.1L2
2.2

If V=U, then h would be a uniform isomorphism from 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 · two levels

13 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