When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:

Let $G$ be a topological group. A character of $G$ is a homomorphism of topological groups from $G$ to the torus $T=\mathbb{R}/\mathbb{Z}$. Here, the torus $T$ has the induced topology from the usual topology of the real line $\mathbb{R}$.

I found the same definition on Wikipedia. So, I think this is a standard definition of character.

But, in a lot of modern articles, it seems to me that characters are defined as follows:

Let $G$ be a topological group. A character of $G$ is a homomorphism of topological groups from $G$ to the discrete additive group $\mathbb{Q}/\mathbb{Z}$.

Clearly, these two definitions cannot be the same for all topological groups.

However, if $G$ is a (discrete) finite group, then the two definitions agree.

Question:

What kind of conditions on a topological group $G$ one needs in order to identify the two definitions of character?

If $G$ is a "profinite group", then, do the two definitions agree? If the answer is yes, then how can one prove it?

Please give me any advice.

# Later

I found a way to answer the second question. One only needs to show the following: for any profinite group $G$ and any continuous homomorphism $f:G \to T$, the image of $f$ is finite. This statement can be shown as follows. The torus $T$ has an open neighborhood $U$ of $0$ which contains no non-tirivial subgroup of $T$. Since $G$ is profinite, there exists an open normal subgroup $H$ of $G$ satisfying $H \subset f^{-1}(U)$. This implies $H \subset Ker(f)$. So, the map $f$ factors through the finite group $G/H$. This concludes that the image of $f$ is finite.

Hence the two definitions of character agree for any profinite group.