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A(Ce abelian group additive group additive reals algebraic central subgroup circle group clearly closure clusterpoint compact boundary compact connected group compact normal subgroups compact open subgroup compact set compact space compact topological group congruence relation continuous mapping converging cosets modulo countable cyclic group dense subgroup direct product division rings finishes the proof finite dimensional finite number G contains group H group with compact group with zero Hausdorff hence homomorphism identity inner automorphisms intersection Lemma Lie group limit of Lie locally compact connected locally compact group locally compact topological maximal compact open maximal compact subgroup maximal subgroup multiplicative group multiplicative semigroup natural numbers non-negative reals normal in G one-parameter group one-parameter subgroup open neighborhood positive reals projective limit Proposition prove quotient space real numbers reals under multiplication sequence structure subgroup G subgroup of G subset theorem topological semigroup totally disconnected trivial