TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every countably based uniformity is generated by one pseudometric, which is a metric exactly when the uniformity is separated

Statement

Every countably based uniformity is generated by one pseudometric. That pseudometric is a metric exactly when the uniformity is separated.

Facts & Assumptions

Given: A countably based uniformity $\mathcal U$ .

[L1]

In ZF it has a decreasing normal symmetric base (A countable entourage base can be replaced in ZF by a decreasing symmetric base whose next triple composite lies in the preceding member).

[L2]

A normal sequence yields a pseudometric with balls cofinal in the sequence (A normal sequence of entourages yields a uniformly continuous pseudometric with controlled dyadic balls).

[L3]

A metric uniformity is separated (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, Separated uniformity: the intersection of all entourages is the diagonal), and a pseudometric is a metric exactly when its zero pairs are diagonal (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

Proof

technique · constructive

1.1

Apply [L1] and [L2] to obtain a pseudometric $p$ whose dyadic balls are cofinal in $\mathcal U$ .

L1L2construct

2.1

Cofinality means that the uniformity generated by $p$ is exactly $\mathcal U$ .

step 1.1

3.1

The zero pairs of $p$ are the intersection of its dyadic entourages, so they are diagonal exactly when $\mathcal U$ is separated; by [L3] this is exactly when $p$ is a metric.

step 2.1L3

4.1

This proves both assertions.

step 2.1step 3.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

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Sources

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