Uniform Structures on Topological Groups and Their Quotients

axiom of countability bijective Cauchy filter Cauchy filter basis Cauchy sequence Čech-complete Clearly coarser coincides compact subset complete topological group condition Consequently Definition Let discrete topology discrete uniformity equivalence relation Example Let Exercise exists final group topology final uniformity finite subset form a basis group and G group and let group completion Hausdorff topological group Hausdorff uniform hence homomorphism implies induced inframetrizable initial uniformity left invariant Lemma Let let G locally compact M₁ metrizable Moreover neighborhood neutral subgroup normal subgroup obtain operation precompact precompact subset product topology PROOF Let Proposition Let prove pseudometrizable quotient map quotient space quotient topology R-Cauchy resp satisfies semidirect product SIN-group statements are equivalent subgroup G surjective Theorem Let topological semidirect product topological space topologically isomorphic U₁ uniform equivalence uniform space uniformly continuous usual topology V₁ whence X₁ Y₁