TITLE: A possible mistake in Walter Rudin, "Fourier analysis on groups"
QUESTION [16 upvotes]: I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):

Suppose $E$ is a coset in $\Gamma_2$ and $\alpha$ is an affine map of $E$ into $\Gamma_1$. THen $\alpha$ can be extended to an affine map of the closure $\overline{E}$ of $E$, and $\alpha(\overline{E})$ is a closed coset in $\Gamma_1$

I think $\alpha(\overline{E})$ is not necessary a closed coset in $\Gamma_1$. The book just states that it is a corollary of the uniform continuity of $\alpha$ but consider $E = \Gamma_2 = \mathbb{Q}$ with discrete topology, $\Gamma_1 = \mathbb{R}$ and $\alpha$ is identity map. In that case, $\overline{E} = E$ and the image is not closed in $\mathbb{R}$.
Edit: Since nobody proved the statement or pointed out where my example is wrong I'm going to assume the lemma is indeed incorrect.

REPLY [6 votes]: [This is just moving my comment to an answer so that the question does not keep showing up as unanswered.]
The discussions above seem to demonstrate that the OP is correct, and that this part of the lemma is incorrect. According to my Canadian sources, it seems that this error has been noticed tacitly by other people: see the remarks before and after Lemma 1.3 in:

M. Ilie, N. Spronk. Completely bounded homomorphisms of the Fourier algebras, J. Funct. Anal. 225 (2005) no. 2, 480–499