LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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.

The minimal Cauchy filters associated to points define a uniformly continuous dense canonical map

Statement

The map $η : X \to \hat{X}$ sending $x$ to the minimal Cauchy filter associated to its principal filter is uniformly continuous and has dense image. For every $x \in X$ , every member of $η (x)$ contains $x$ .

Facts & Assumptions

Given: A uniform space $X$ and its minimal-Cauchy-filter space $\hat{X}$ .

[L1]

Principal filters are Cauchy and have associated minimal Cauchy filters (Every Cauchy filter canonically determines a unique minimal Cauchy filter coarser than it).

[L2]

The standard relations are entourages on $\hat{X}$ (The standard entourages on minimal Cauchy filters form a separated uniformity).

[L3]

Entourage balls describe the induced topology and density is closure equal to the whole space (The sets containing an entourage ball about each of their points form a topology, Interior, closure, boundary, exterior, derived set and isolated point in a topological space).

[L4]

Symmetric entourages with prescribed finite-composite control may be chosen inside any entourage (Every uniformity has a base of symmetric entourages).

Proof

technique · constructive

1.1

Define $η (x)$ to be the minimal filter associated to the principal filter $P_{x}$ at $x$ . Since $η (x) = m (P_{x}) \subseteq P_{x}$ , every member of $η (x)$ contains $x$ .

L1construct

1.2

Let $\hat{E} [F]$ be a basic neighbourhood. Choose a symmetric $D$ with $D^{\circ 2} \subseteq E$ , a $D$ -small $A \in F$ , and $a \in A$ . The point filter $η (a)$ contains $D [a]$ , and $D [a] \times A \subseteq D^{\circ 2} \subseteq E$ , so $η (a) \in \hat{E} [F]$ . Thus every basic neighbourhood meets $η [X]$ .

L1L2L3L4choose

2.1

Given a target basic entourage $\hat{E}$ , choose a symmetric $D$ with $D^{\circ 3} \subseteq E$ . If $(x, y) \in D$ , then $D [x] \in η (x)$ and $D [y] \in η (y)$ , while $D [x] \times D [y] \subseteq D^{\circ 3} \subseteq E$ . Hence $(η (x), η (y)) \in \hat{E}$ , which proves uniform continuity.

step 1.1L2L4

3.1

Thus every neighbourhood meets $η [X]$ , so its closure is all of $\hat{X}$ and the image is dense.

step 1.2L3discharge-construct∎

Depends on

Used by

Every uniform space has a Hausdorff completion with dense canonical image, and the canonical map is a uniform embedding exactly when the original uniformity is separated Theorem

Dependency tree · two levels

13 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)
J. H. V. Hunt, Boletín de la Sociedad Matemática Mexicana 34 (1989), 11–21 (standard reference, not scraped)