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Another example: the diagram in the German page says that <math>\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n\subset\mathbf{\Delta}^1_{n+1}</math>, which is equivalent to <math>\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n\subset\mathbf{\Sigma}^1_{n+1}</math>. This means that every <math>\mathbf{\Sigma}^1_n</math> set and every <math>\mathbf{\Pi}^1_n</math> is the projection of a <math>\mathbf{\Pi}^1_n</math> set; while the latter implication is trivially true, I have no idea about the former even with ''n'' = 1. The inclusion gives us the inclusion of σ-algebras <math>\mathbf{\Delta}^1_1\subset\sigma(\mathbf{\Sigma}^1_1)\subset\mathbf{\Delta}^1_2\subset\cdots\subset\mathbf{P}</math>. Perhaps in fact each inclusion is strict in eveyy uncountable Polish space? (We know that this is true for the first inclusion as stated [https://webusers.imj-prg.fr/~dominique.lecomte/Chapitres/6-Analytic%20and%20co-analytic%20sets.pdf here], of course [https://math.stackexchange.com/q/3638461 some choice may be needed]; [https://mathoverflow.net/q/425003 this post] may have addressed the second, although I don't know if it would work for eveyy uncountable Polish space.)
What's more, the inclusion <math>\mathbf{\Pi}^1_n\subset\mathbf{\Delta}^1_{n+1}</math> tells us that <math>\mathbf{\Sigma}^1_n</math> sets are precisely the projections of <math>\mathbf{\Delta}^1_n</math> sets, so <math>\mathbf{\Delta}^1_n</math> sets are precisely those sets such that themselves as well as their projections are all <math>\mathbf{\Delta}^1_n</math> sets: by definition <math>\mathbf{\Delta}^1_n\subset\mathbf{\Sigma}^1_n</math>, and the projections of <math>\mathbf{\Sigma}^1_n</math> sets are also in <math>\mathbf{\Sigma}^1_n</math>
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