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Assuming the ultrafilter lemma, a uniform space is compact if and only if it is complete and totally bounded
Statement
Assume the ultrafilter lemma. A uniform space is compact if and only if it is complete and totally bounded.
Facts & Assumptions
Given: A uniform space and the ultrafilter lemma.
Compact uniform spaces are complete (Every compact uniform space is complete).
Compact uniform spaces are totally bounded (Every compact uniform space is totally bounded).
Under the ultrafilter lemma, complete totally bounded uniform spaces are compact (Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact).
Proof
Compactness implies completeness and total boundedness by [L1] and [L2].
Completeness together with total boundedness implies compactness by [L3].
The two implications prove the equivalence under the stated assumption.
Depends on
Used by
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Sources
- J. Wodzicki, Uniform Structure (standard reference, not scraped)
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)