Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be said? In particular, is the following true?



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> Every abelian Hausdorff topological group has a nontrivial continuous (unitary) character
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I believe this to be false, but cannot find a concrete example.



A Hausdorff abelian group with no character?
The infinite-dimensional sphere $S^\infty$ (the evident colimit of finite-dimensional spheres) admits a topological group structure whose underlying abelian group is a torsion group (in fact a structure of $\mathbb{F}\_2$-vector space). See this answer here: <https://mathoverflow.net/a/43047/2926>. 


If $\phi: S^\infty \to S^1$ is a continuous character, then $\phi(S^\infty)$ is evidently a connected torsion subgroup of $S^1$, which can only be trivial since the torsion subgroup $\mathbb{Q}/\mathbb{Z}$ of $S^1$ is totally disconnected. 


