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The Samuel completion and, when compactifying, the Samuel compactification
Definition
A Samuel completion of is a Hausdorff completion of its Samuel uniformity in the sense of A Hausdorff completion of a uniform space and its canonical dense map. Such a completion exists by Every uniform space has a Hausdorff completion with dense canonical image, and the canonical map is a uniform embedding exactly when the original uniformity is separated, but its canonical map need not be injective.
Regard with its original induced topology. A Samuel completion is a Samuel compactification if and only if the same map makes a compactification in the sense of A Hausdorff compactification as a dense embedding into a compact Hausdorff space. In particular, this requires to be compact Hausdorff and to be an embedding with dense image; it is not used merely for a Hausdorff completion of a nonseparated uniform space.
Depends on
- The Samuel uniformity generated by bounded uniformly continuous functions
- A Hausdorff completion of a uniform space and its canonical dense map
- Every uniform space has a Hausdorff completion with dense canonical image, and the canonical map is a uniform embedding exactly when the original uniformity is separated
- A Hausdorff compactification as a dense embedding into a compact Hausdorff space
Used by
- Samuel compactifications are unique up to the unique isomorphism fixing the original space Corollary
- Under dependent choice the Samuel completion of a separated totally bounded space is its uniform completion; under the ultrafilter lemma it is compact Corollary
- The Samuel reflection of a nonempty indiscrete uniform space is a singleton Example
- Under dependent choice and the ultrafilter lemma, uniformly continuous maps to compact Hausdorff spaces extend uniquely over the Samuel compactification Theorem
- Under the ultrafilter lemma the Samuel completion is compact, and under dependent choice plus the ultrafilter lemma it compactifies every separated uniform space Theorem
Dependency tree · two levels
23 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
- Garrido and Meroño, The Samuel realcompactification (standard reference, not scraped)
- M. Megrelishvili, Samuel and Smirnov compactifications (standard reference, not scraped)