The Pontryagin dual with the compact-open topology

Let $G$ be an abelian topological group (Topological group: multiplication and inversion are continuous) written additively, and let $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ be the multiplicative unit circle (The multiplicative unit circle is a compact metrizable topological abelian group).

A character of $G$ is a continuous group homomorphism $\gamma :G\to \mathbb{T}$ (Monoid homomorphism and group homomorphism, Continuity of a map of topological spaces at a point and globally). The Pontryagin dual of $G$ is $\widehat{G}:={\mathrm{Hom}}_{cts}(G,\mathbb{T}):=\{\gamma :G\to \mathbb{T}:\gamma \text{ is a continuous group homomorphism}\},$ the set of characters, equipped with:

1. Pointwise multiplication. For ${\gamma }_1,{\gamma }_2\in \widehat{G}$ the product is $({\gamma }_1{\gamma }_2)(x):={\gamma }_1(x){\gamma }_2(x)$ for every $x\in G$, with the constant character $x\mapsto 1$ as its identity and $x\mapsto \gamma (x{)}^{-1}$ as the inverse of $\gamma$. With these operations the set of characters is a group, and it is abelian; this and the continuity of the two operations are proved in the next item, so no separate well-definedness obligation is left open here.
2. Compact-open topology. The topology is the compact-open topology inherited from $C(G,\mathbb{T})$ (The compact-open topology on $C(X,Y)$ for arbitrary topological spaces), that is the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace) for the subbasis $S(K,V):=\{\gamma \in \widehat{G}:\gamma [K]\subseteq V\},K\subseteq G\text{ compact},V\subseteq \mathbb{T}\text{ open}\mathrm{.}$

The evaluation pairing is written $⟨x,\gamma ⟩:=\gamma (x)\in \mathbb{T},x\in G,\gamma \in \widehat{G}\mathrm{.}$

The dual is written multiplicatively, so products of characters are written ${\gamma }_1{\gamma }_2$ and the identity is written $1$. Through the topological group isomorphism $\epsilon :\mathbb{R}\mathrm{/}\mathbb{Z}\to \mathbb{T}$ of The multiplicative unit circle is a compact metrizable topological abelian group, characters may equivalently be viewed as continuous homomorphisms into the published circle $\mathbb{R}\mathrm{/}\mathbb{Z}$; all statements below use the multiplicative circle $\mathbb{T}$ and the compact-open subbasis displayed above.