I am wondering the question in the title: let $X$ be uncountable Polish. Consider the standard Borel structure on $X$; that is, $\mathbf{\Sigma}_1^0(X)$ are the open sets, etc.. Is it true that, with respect to some notion of computability, every $\mathbf{\Sigma}_1^1(X)$-set is $\Sigma_1^1(X)$ with respect to some oracle? I am aware this is true for $X = 2^{\omega}$, but since, for instance, coanalytic sets are not preserved under Borel isomorphism, I’m not sure whether this extends to arbitrary Polish spaces. If someone has a reference text, that would be helpful as well!
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2$\begingroup$ I like this word: ogithface . $\endgroup$– GEdgarDec 14, 2021 at 1:18
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$\begingroup$ @GEdgar I was watching Shrek earlier… must’ve still been on my mind! $\endgroup$– MacRanceDec 14, 2021 at 5:49
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$\begingroup$ @MacRance lmao nice one $\endgroup$– p_squareDec 14, 2021 at 5:54
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$\begingroup$ How is $\Sigma^1_1(X)^A$ defined for an oracle $A$, for a general Polish space $X$? (I'm making up this notation here, I don't know what's actually used.) $\endgroup$– Noah SchweberDec 14, 2021 at 6:09
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$\begingroup$ @NoahSchweber that is a very good question, and one that I have no answer to to be honest — this should probably form part of my question, I left it out in the assumption there’s an agreed upon notion I’m just unaware of. $\endgroup$– MacRanceDec 14, 2021 at 7:11
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