TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Assuming dependent choice, every entourage uniformity is generated by a gauge of uniformly continuous pseudometrics

Statement

Assuming dependent choice, every entourage uniformity is generated by a gauge of uniformly continuous pseudometrics.

Facts & Assumptions

Given: An entourage uniformity $\mathcal U$ and dependent choice.

[L1]

For every entourage, dependent choice supplies a normal sequence whose first nontrivial member is contained in that entourage (Assuming dependent choice, every entourage admits a normal symmetric sequence subordinate to it).

[L2]

Such a sequence yields a uniformly continuous pseudometric whose dyadic balls lie between consecutive entourages (A normal sequence of entourages yields a uniformly continuous pseudometric with controlled dyadic balls).

[L3]

A gauge generates the filter based on finite simultaneous pseudometric balls (A gauge of pseudometrics and, on a nonempty set, the uniformity it generates).

Proof

technique · constructive

1.1

Let $\mathcal P$ be the set of all pseudometrics obtained by applying [L2] to normal sequences of [L1] subordinate to some entourage. This definition makes no simultaneous choice. Every $p\in\mathcal P$ is uniformly continuous for $\mathcal U$ , and for each entourage $E$ , [L1] and [L2] ensure that at least one $p\in\mathcal P$ has a positive-radius $p$ -ball contained in $E$ .

L1L2construct

2.1

For each $p\in\mathcal P$ and each positive radius, its ball is an original entourage by step 1.1. Hence every finite intersection defining a basic entourage of the gauge belongs to $\mathcal U$ .

step 1.1L3

2.2

Conversely every original entourage $E$ contains a positive-radius ball for at least one $p\in\mathcal P$ by step 1.1, so it belongs to the gauge uniformity.

step 1.1L3

3.1

The two uniformities contain one another and are equal.

step 2.1step 2.2discharge-construct∎

Depends on

Used by

On a nonempty set, entourages and uniform covers give equivalent definitions of a uniform space in ZF, and under dependent choice they are also equivalent to gauges of pseudometrics Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 90 results over 20 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. Kunzinger, General Topology (standard reference, not scraped)