Characters of $C(X)$

Question

I showed that the character space $\Omega (\ell^1 (\mathbb Z))$ is homeomorphic to $S^1$.

Now I am wondering if there is a similar identification for $C(X)$ where $X$ is compact Hausdorff with the sup norm?

Answer 1

Yes. The character space of $C(X)$ is homemorphic to $X$.

Proving this is a good exercise, but if you need a hint, the key ideas are contained in this question.