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A Hausdorff completion of a uniform space and its canonical dense map
Definition
A Hausdorff completion of a uniform space is a complete separated uniform space together with a map satisfying both of the following conditions.
- The image is dense: (Interior, closure, boundary, exterior, derived set and isolated point in a topological space).
- The original uniformity is exactly the uniformity pulled back along : for every , , and for every there is with .
The first half of the second condition is uniform continuity (Uniformly continuous map between uniform spaces); the second half prevents the completion map from discarding any of the original uniform structure. The map is not required to be injective. It is a uniform embedding (Uniform embedding and uniform isomorphism) exactly when it is injective, and this is the usual completion of a separated uniform space.
Depends on
- Complete uniform space: every Cauchy filter converges
- Separated uniformity: the intersection of all entourages is the diagonal
- Uniform embedding and uniform isomorphism
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Uniformly continuous map between uniform spaces
Used by
- 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 compactification map need not be a uniform embedding for the original uniformity Counterexample
- The Samuel completion and, when compactifying, the Samuel compactification Definition
- The Samuel reflection of a nonempty indiscrete uniform space is a singleton Example
- Under dependent choice and the ultrafilter lemma, the Samuel compactification of the open unit interval is the closed unit interval Example
- 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 Theorem
- 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 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
12 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)
- Encyclopedia of Mathematics, Complete uniform space (standard reference, not scraped)