Dual group of $\mathbb Z$

Question

We know $\hat{\mathbb Z}=\mathbb T$ and the map $\alpha\longmapsto\chi_{\alpha}$ is an isomorphism of $\mathbb T$ on to the character group of $\mathbb Z$, but I can't prove this map is continuous? I don't know what topology on $\hat{\mathbb Z}$ is defined.

Answer 1

In Pontryagin duality, the gual group $\widehat G$ is the group of all continuous group homomorphisms $G\to\mathbb T$ (characters of $G$) with the compact-open topology.

In the case you are interested in we take $\mathbb Z$ with the discrete topology. This means that compact subsets are precisely the finite subsets and the compact-open topology is precisely the product topology.

You ask about continuity of the map from $\mathbb T$ to $\widehat{\mathbb Z}$ defined by $\alpha\mapsto\chi_\alpha$, where $\chi_\alpha(z)=\alpha^z$.

Since we take the topology induced by the product topology on $\widehat{\mathbb Z}$, it is sufficient to check that for each $\mathbb Z$ the map $\alpha\mapsto \chi_\alpha(z)$ is continuous. (This is the composition of our map with the projection on the $z$-th coordinate.) But this is simply the continuity of the function $\mathbb T\mapsto\mathbb T$ given by $\alpha\mapsto\alpha^z$ (for an integer $z$). Continuity of this function is well-known. (It is a restriction of continuous map $\alpha\mapsto\alpha^z$ from $\mathbb C$ to $\mathbb C$.)

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(I should point out that this does not finishes the proof that $\mathbb T$ and $\widehat{\mathbb Z}$ are isomorphic. We should check whether the map is bijective, the continuity of the inverse map and we should also check whether these maps are group homomorphisms. But the question specifically asks about continuity of the above map, so I discussed only this part in my answer.)