I want to know if anybody has some clue as why a compact metrizable abelian group has enough characters to separate points (a character on a topological group is a continuous homomorphism from the group to the circle group). Separate points means that if g is not 0, then there exists a character f such that f(g) is not 0.
2026-03-27 23:29:15.1774654155
Showing that compact metrizable abelian group has enough characters
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This is a well-known result from the theory of locally compact abelian groups. But I think that its proof should be complicated. For instance, in [Pon] this is a corollary from $\S$34.C, with a long full proof, which is based on irreducible matrix representations of compacts abelian groups, which are bases on integration with respect to invariant measures on these groups.
References
[Pon] Lev S. Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian).