I have the next problem:
Let $G$ be a compact abelian group and $H$ the group of continuous characters on G (i.e. continuous homomorphism from G to the circle $T$) with the topology of pointwise convergence. Is $H$ a discrete space?
Note: The pointwise convergence topology in $H$ is the subspace topology of $H$ seen as a subspace of $T^G$.
I think the question is false assuming $G=T$. I already showed that the continuous homomorphism from $T$ to $T$ are functions of the form $f(z)=z^n$ for some $n \in \mathbb{N}$.
Now, I'm trying to prove (without success yet) that the identity function $g=1$ can't be open in $H$, which is equivalent to show that for any finite subset $\{z_1,\ldots, z_n\}\subseteq T$ and $\{U_1,\ldots, U_n\}$ open neighbors of $1$ on $T$, exists $n\in \mathbb{N}$ such that $f^n(z_i)\in U_i \quad \forall i\leq n$.
Any suggestion is appreciated, Thanks!!