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A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).
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Dense cyclic subgroup
How about the infinite cyclic group itself with the discrete topology? Or p-adic integers?
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Accepted
Two Definitions of "Character" of topological groups
I am assuming all groups we are talking about are locally compact and commutative.
The two definitions indeed do ageree on profinite groups. To prove it, you have to check that the functors $Hom(-,\m …