LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Samuel function pseudometrics generate a uniformity coarser than the original one

Statement

For every $f\in\mathcal F_{\mathcal U}$ , the function $p_f(x,y)=|f(x)-f(y)|$ of The Samuel uniformity generated by bounded uniformly continuous functions is a pseudometric. Every basic Samuel entourage is an entourage of $\mathcal U$ ; hence $\mathcal U_S\subseteq\mathcal U$ . Moreover the gauge obtained from all bounded real-valued uniformly continuous functions generates the same uniformity as $\mathcal U_S$ .

Facts & Assumptions

Given: A uniform space $(X,\mathcal U)$ , a uniformly continuous $f:X\to[0,1]$ , and positive reals $\varepsilon$ and $M$ .

[L1]

For real numbers, $|u|\ge0$ , $|u|=0$ exactly when $u=0$ , and $|-u|=|u|$ (Basic properties of the absolute value).

[L2]

The real triangle inequality is $|u+v|\le|u|+|v|$ (The triangle inequality).

[L3]

The sets $\{(s,t):|s-t|<\varepsilon\}$ generate the metric uniformity of $[0,1]$ , and metric uniform continuity is the corresponding epsilon-delta condition (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).

Proof

technique · direct

1.1

The diagonal and symmetry axioms for $p_f$ follow from [L1], while $p_f(x,z)\le p_f(x,y)+p_f(y,z)$ follows by applying [L2] to $f(x)-f(z)=(f(x)-f(y))+(f(y)-f(z))$ ; thus $p_f$ is a pseudometric.

L1L2

1.2

For every $\varepsilon>0$ , uniform continuity of $f$ and [L3] give an entourage $U$ of $\mathcal U$ with $(x,y)\in U$ implying $p_f(x,y)<\varepsilon$ .

1.3

If $g:X\to\mathbb R$ is uniformly continuous with $|g|\le M$ , then for $M>0$ the map $h=(g+M)/(2M)$ is $[0,1]$ -valued and uniformly continuous: for a target tolerance $\delta>0$ , use uniform continuity of $g$ with tolerance $2M\delta$ . Moreover $p_g=2M p_h$ ; if $M=0$ , $g$ is constant. Thus the two gauges have the same basic entourages.

construct

2.1

For a basic Samuel entourage, step 1.2 gives an original entourage inside each of its finitely many coordinate balls; their finite intersection is therefore an original entourage contained in the basic Samuel entourage. Hence every basic Samuel entourage belongs to $\mathcal U$ , and so does the generated filter.

step 1.2

3.1

Steps 1.1, 2.1, and 1.3 prove all assertions.

step 1.1step 2.1step 1.3∎

Depends on

Used by

Cited to discharge well-definedness by The Samuel uniformity generated by bounded uniformly continuous functions.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 42 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

Garrido and Meroño, The Samuel realcompactification (standard reference, not scraped)
J. Wodzicki, Uniform Structure (standard reference, not scraped)