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Under dependent choice the Samuel completion of a separated totally bounded space is its uniform completion; under the ultrafilter lemma it is compact
Statement
Assume dependent choice. For a separated totally bounded uniform space , every Samuel completion is, up to the unique uniform isomorphism fixing , a Hausdorff completion of the original uniformity. Assume also the ultrafilter lemma. This common completion is compact.
Facts & Assumptions
Given: A separated totally bounded uniform space , dependent choice, and, for compactness, the ultrafilter lemma.
For a totally bounded uniform space, dependent choice makes the original and Samuel uniformities equal (Assuming dependent choice, a totally bounded uniformity equals its Samuel uniformity).
Hausdorff completions are unique up to the unique uniform isomorphism commuting with their canonical maps (Every uniformly continuous map into a complete Hausdorff uniform space extends uniquely across the Hausdorff completion; consequently completions are unique up to a unique uniform isomorphism).
A Hausdorff completion has a dense uniformly continuous canonical map and complete target; a dense uniformly continuous image of a totally bounded space is totally bounded, and under the ultrafilter lemma a complete totally bounded uniform space is compact (A Hausdorff completion of a uniform space and its canonical dense map, Total boundedness passes to a uniform space with a dense uniformly continuous image, Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact).
Proof
By [L1], a Samuel completion is a Hausdorff completion of the original uniformity.
The uniqueness theorem [L2] identifies it with every other Hausdorff completion by the unique uniform isomorphism fixing .
Its dense canonical image and [L3] make it totally bounded, while a Hausdorff completion is complete; hence [L3] makes it compact under the ultrafilter lemma.
This proves the DC identification and the separately qualified compactness assertion.
Depends on
- The Samuel completion and, when compactifying, the Samuel compactification
- A Hausdorff completion of a uniform space and its canonical dense map
- Assuming dependent choice, a totally bounded uniformity equals its Samuel uniformity
- Total boundedness passes to a uniform space with a dense uniformly continuous image
- Every uniformly continuous map into a complete Hausdorff uniform space extends uniquely across the Hausdorff completion; consequently completions are unique up to a unique uniform isomorphism
- Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact
Used by
- 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
- Under dependent choice and the ultrafilter lemma, the Samuel compactification of the open unit interval is the closed unit interval Example
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 79 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)