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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 be a nonempty compact Hausdorff space and give it its unique compatible uniformity. Its Samuel compactification is then uniformly isomorphic over to itself, and the identity realizes the corresponding ordinary completion.
Facts & Assumptions
Given: A nonempty compact Hausdorff space with its unique compatible uniformity.
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).
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
By [L1], the identity is a Hausdorff completion: it is a uniform embedding, its image is dense, and its target is complete.
Apply [L2]: every Samuel completion is uniformly isomorphic over to the identity completion of step 1.1, and under the stated choice principles it is the Samuel compactification.
Thus no point is added; the singleton case is included.
Depends on
- Under dependent choice the Samuel completion of a separated totally bounded space is its uniform completion; under the ultrafilter lemma it is compact
- Every compact uniform space is complete
- Every compact uniform space is 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
Used by
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Sources
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)