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Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact
Statement
Assume the ultrafilter lemma. Every complete and totally bounded uniform space is compact.
Facts & Assumptions
Given: A complete, totally bounded uniform space and the ultrafilter lemma.
Every ultrafilter on a totally bounded uniform space is Cauchy (Every ultrafilter on a totally bounded uniform space is Cauchy).
Completeness makes every Cauchy filter converge (Complete uniform space: every Cauchy filter converges).
Assuming the ultrafilter lemma, a topological space is compact if and only if every ultrafilter converges (Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging).
Proof
Let be an ultrafilter on . It is Cauchy by [L1].
Completeness makes converge by [L2].
Every ultrafilter converges, so is compact by [L3], under the stated ultrafilter-lemma assumption.
Depends on
- Every ultrafilter on a totally bounded uniform space is Cauchy
- Complete uniform space: every Cauchy filter converges
- Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging
Used by
- Assuming the ultrafilter lemma, a uniform space is compact if and only if it is complete and totally bounded 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
- 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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 41 results over 7 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
- J. Wodzicki, Uniform Structure (standard reference, not scraped)
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)