Alphabeta MathHow statement and proof provenance work
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Every compact uniform space is complete
Statement
Every compact uniform space is complete.
Facts & Assumptions
Given: A compact uniform space and a Cauchy filter on it.
Compactness is equivalent to the finite-intersection property for closed families (A space is compact exactly when every family of closed subsets with the finite intersection property has nonempty intersection, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).
A Cauchy filter with a cluster point converges to that point (A Cauchy filter with a cluster point converges to that point).
Completeness means convergence of every Cauchy filter (Complete uniform space: every Cauchy filter converges).
Proof
The closures of the members of have the finite-intersection property, because finite intersections of filter members are nonempty and lie in the corresponding intersections of closures.
Compactness gives ; every neighbourhood of therefore meets every , so is a cluster point of .
By [L2] the filter converges, and since it was arbitrary is complete by [L3].
Depends on
- Complete uniform space: every Cauchy filter converges
- A Cauchy filter with a cluster point converges to that point
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A space is compact exactly when every family of closed subsets with the finite intersection property has nonempty intersection
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 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
- Under dependent choice and the ultrafilter lemma, uniformly continuous maps to compact Hausdorff spaces extend uniquely over the Samuel compactification Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 41 results over 10 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)
- Encyclopedia of Mathematics, Complete uniform space (standard reference, not scraped)