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I am looking for an example of a unit circle subset that is $F_{\sigma\delta}$ but not $F_\sigma$. The task is connected with studying convergence of power series on the boundary of the convergence set, so this is why being a subset of circle is needed.

Is $\{z\in\mathbb{C}:|z|=1\}- \{z\in\mathbb{C}:z=e^{i\pi\phi}, \phi\in\mathbb{Q}\}$ a good example?

maq
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Yes, that seems right. It's not $F_{\sigma}$ by the argument given in answer to this question and it's $F_{\sigma\delta}$ because it equals $$\bigcap_{q\in \mathbf Q}S^1\setminus\{e^{i\pi q}\}$$

saulspatz
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  • I am having a little trouble with formal proof and applying the same argument in that case, since using the complement of my set is not so easy - at least to me. Could you elaborate something more on this subject, please? – maq Apr 24 '18 at 17:42
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    I'm uncertain what your difficulty is. Why doesn't the proof translate almost word for word? If $S^1\setminus{e^{i\pi\phi}|\phi\in\mathbb Q}$ is $F_{\sigma}$ then ${e^{i\pi\phi}|\phi\in\mathbb Q}$ is $G_{\delta}$ etc. – saulspatz Apr 24 '18 at 18:13