ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

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Under dependent choice and the ultrafilter lemma, the Samuel compactification of a nonempty compact Hausdorff space adds no points up to unique uniform isomorphism

Example

Assume dependent choice and the ultrafilter lemma. Let $K$ be a nonempty compact Hausdorff space and give it its unique compatible uniformity. Its Samuel compactification is then uniformly isomorphic over $K$ to $K$ itself, and the identity ${id}_{K}$ realizes the corresponding ordinary completion.

Facts & Assumptions

Given: A nonempty compact Hausdorff space $K$ with its unique compatible uniformity.

[L1]

The compatible uniformity induces the given Hausdorff topology and is therefore separated; compact uniform spaces are complete and totally bounded (A nonempty compact Hausdorff space carries exactly one compatible uniformity, A uniformity is separated if and only if its induced topology is Hausdorff, Every compact uniform space is complete, Every compact uniform space is totally bounded).

[L2]

Under dependent choice, every Samuel completion of a separated totally bounded space is, up to the unique uniform isomorphism fixing that space, its ordinary uniform completion; under the ultrafilter lemma this common completion is compact (Under dependent choice the Samuel completion of a separated totally bounded space is its uniform completion; under the ultrafilter lemma it is compact).

Verification

technique · direct

1.1

By [L1], the identity $K \to K$ is a Hausdorff completion: it is a uniform embedding, its image is dense, and its target is complete.

2.1

Apply [L2]: every Samuel completion is uniformly isomorphic over $K$ to the identity completion of step 1.1, and under the stated choice principles it is the Samuel compactification.

L2step 1.1

3.1

Thus no point is added; the singleton case is included.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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