The Hopf-Rinow theorem can generalize this theorem to complete Riemann manifolds. Its first proof utilized a Lebesgue chain, and one of its namesakes originally linked it to u
The Hopf-Rinow theorem can generalize this theorem to complete Riemann manifolds. Its first proof utilized a Lebesgue chain, and one of its namesakes originally linked it to uniform continuity. Veblen noted its logical equivalence to the statement that every cut on an ordered field F goes through some element, the Dedekind Cut Axiom. The Weierstrauss property provides equivalence to this statement if every sequence in the set has a convergent subsequence, though this is commonly stated as every cover having a finite subcover when applied to metric spaces, where this theorem holds for completeness and total boundedness. FTP, identify this doubly eponymous theorem which states that a subset of a Euclidean space is closed and bounded if and only if it is compact.
The Hopf-Rinow theorem can generalize this theorem to complete Riemann manifolds.
Heine-Borel-Lebesgue theorem or property [prompt on partial answer]