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

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Assuming dependent choice, the Samuel uniformity induces the original topology

Statement

Assume dependent choice. The topology induced by $U_{S}$ equals the topology induced by $U$ .

Facts & Assumptions

Given: Dependent choice, an original-open set $O \subseteq X$ , and a point $x \in O$ .

[L1]

The Samuel uniformity is coarser than the original uniformity (Samuel function pseudometrics generate a uniformity coarser than the original one).

[L2]

Entourage balls form a neighbourhood base for the induced topology (The sets containing an entourage ball about each of their points form a topology).

[L3]

Under dependent choice, every entourage $U$ has a decreasing normal symmetric sequence with $E_{1} \subseteq U$ (Assuming dependent choice, every entourage admits a normal symmetric sequence subordinate to it, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[L4]

A normal sequence gives a uniformly continuous pseudometric $p$ with ${p \leq 2^{- 2}} \subseteq E_{1}$ (A normal sequence of entourages yields a uniformly continuous pseudometric with controlled dyadic balls).

[L5]

The ball of a Samuel coordinate is a Samuel entourage-ball (The Samuel uniformity generated by bounded uniformly continuous functions).

Proof

technique · constructive

1.1

Since $U_{S} \subseteq U$ , every Samuel-open set is original-open.

L1L2

1.2

Choose $U \in U$ with $U [x] \subseteq O$ , take the sequence of [L3], and take the pseudometric $p$ of [L4]; then ${y : p (x, y) \leq 1 / 4} \subseteq O$ .

L2L3L4

1.3

Put $f (y) = \min {1, 4 p (x, y)}$ . The reverse triangle inequality for a pseudometric and the uniform continuity of $p$ make $f$ uniformly continuous, so $f \in F_{U}$ and $f (x) = 0$ .

L4construct

2.1

The Samuel neighbourhood ${y : ∣ f (y) - f (x) ∣ < 1}$ lies in ${y : p (x, y) < 1 / 4} \subseteq O$ , so every original-open set is Samuel-open.

L5step 1.2step 1.3

3.1

The two inclusions in steps 1.1 and 2.1 give equality of the topologies; for $X = \emptyset$ both are the empty topology.

step 1.1step 2.1discharge-construct∎

Depends on

Used by

Under the ultrafilter lemma the Samuel completion is compact, and under dependent choice plus the ultrafilter lemma it compactifies every separated uniform space Theorem

Dependency tree · two levels

36 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

J. Wodzicki, Uniform Structure (standard reference, not scraped)