LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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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.

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Every compact uniform space is totally bounded

Statement

Every compact uniform space is totally bounded.

Facts & Assumptions

Given: A compact uniform space $X$ and an entourage $E$ .

[L1]

Symmetric entourages form a base (Every uniformity has a base of symmetric entourages).

[L2]

Compactness gives a finite subcover of every open cover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[L3]

Total boundedness asks for a finite family of entourage balls (Totally bounded uniform space).

[L4]

Every entourage ball is a neighbourhood and hence contains an open neighbourhood of its centre (The sets containing an entourage ball about each of their points form a topology).

Proof

technique · direct

1.1

Choose a symmetric $D \subseteq E$ . For each $x \in X$ , let $O_{x}$ be the union of all open subsets of $D [x]$ that contain $x$ . By [L4], $O_{x}$ is an open neighbourhood of $x$ contained in $D [x]$ , and the family $(O_{x})_{x \in X}$ covers $X$ .

L1L4

2.1

Compactness gives finite $F \subseteq X$ with $X = ⋃_{x \in F} O_{x} \subseteq ⋃_{x \in F} D [x]$ .

step 1.1L2

3.1

Since $D [x] \subseteq E [x]$ , the same finite set covers $X$ by $E$ -balls, proving total boundedness by [L3].

step 2.1L3∎

Depends on

Used by

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)
Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)