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

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On a nonempty set, entourage uniformities and uniform-cover structures determine one another

Statement

In ZF, on a nonempty set $X$ , an entourage uniformity determines a uniform-cover structure by the covers ${E [x] : x \in X}$ , and a uniform-cover structure determines an entourage uniformity by the sets $⋃_{V \in V} V \times V$ . These constructions recover the same uniform structure.

Facts & Assumptions

Given: A nonempty set $X$ carrying either an entourage uniformity or a uniform-cover structure.

[L1]

Symmetric entourages form a base and have symmetric square roots (Every uniformity has a base of symmetric entourages, Uniform space in the entourage formulation).

[L2]

Uniform covers are upward closed under coarsening, have common refinements, and have star-refinements (Uniform space in the uniform-cover formulation).

Proof

technique · constructive

1.1L1construct

From an entourage $E$ , form $C_{E} = {E [x] : x \in X}$ . Choose a symmetric entourage $D$ with $D^{\circ 3} \subseteq E$ . If $D [y] \cap D [x] \neq \emptyset$ and $z \in D [y]$ , symmetry and a point in the intersection give $(x, z) \in D^{\circ 3} \subseteq E$ . Hence the star of $D [x]$ in $C_{D}$ lies in $E [x]$ , so $C_{D}$ star-refines $C_{E}$ .

1.2L2construct

From a uniform cover $V$ , form $E_{V} = ⋃_{V \in V} V \times V$ . It contains the nonempty diagonal, so it is nonempty. A star-refinement $W$ has $E_{W} \circ E_{W} \subseteq E_{V}$ , while common refinements and coarsenings give the remaining filter axioms.

2.1step 1.1L1L2

Declare a cover uniform when it is coarser than some $C_{E}$ . Intersections of entourages give common refinements, enlargement of an entourage gives coarsening, and step 1.1 gives star-refinements. Hence these covers satisfy the uniform-cover axioms.

2.2L1step 1.1step 1.2

Start with an entourage uniformity. For symmetric $D$ , $D \subseteq E_{C_{D}} \subseteq D^{- 1} \circ D = D^{\circ 2} .$ The first inclusion uses the diagonal, and the second follows because two points in one $D$ -ball are $D^{- 1} \circ D$ -related. Taking a symmetric square root inside any prescribed entourage shows that the recovered entourage filter is exactly the original one.

2.3L2step 1.2

Start instead with a uniform-cover structure. The $E_{V}$ -ball at $x$ is $E_{V} [x] = St (x, V),$ the union of the members of $V$ containing $x$ . Thus $V$ refines $C_{E_{V}}$ , so the latter is uniform by coarsening. Conversely, if $W$ star-refines $V$ , then for any $x$ and any $W_{0} \in W$ containing $x$ , $St (x, W) \subseteq St (W_{0}, W)$ , which lies in some member of $V$ . Therefore $C_{E_{W}}$ refines $V$ . The recovered cover structure is exactly the original one.

3.1step 2.2step 2.3discharge-construct∎

Steps 2.2 and 2.3 prove that the two constructions are mutually inverse at the level of generated structures.

On a nonempty set, entourage uniformities and uniform-cover structures determine one another

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources