LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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Samuel function pseudometrics generate a uniformity coarser than the original one

Statement

For every $f \in F_{U}$ , 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 $U$ ; hence $U_{S} \subseteq U$ . Moreover the gauge obtained from all bounded real-valued uniformly continuous functions generates the same uniformity as $U_{S}$ .

Facts & Assumptions

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

[L1]

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

[L2]

The real triangle inequality is $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ (The triangle inequality).

[L3]

The sets ${(s, t) : ∣ s - t ∣ < ε}$ 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 $p_{f} (x, z) \leq p_{f} (x, y) + p_{f} (y, z)$ 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 $ε > 0$ , uniform continuity of $f$ and [L3] give an entourage $U$ of $U$ with $(x, y) \in U$ implying $p_{f} (x, y) < ε$ .

1.3

If $g : X \to R$ is uniformly continuous with $∣ g ∣ \leq M$ , then for $M > 0$ the map $h = (g + M) / (2 M)$ is $[0, 1]$ -valued and uniformly continuous: for a target tolerance $δ > 0$ , use uniform continuity of $g$ with tolerance $2 M δ$ . Moreover $p_{g} = 2 M p_{h}$ ; 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 $U$ , 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 · two levels

22 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

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