# gn.common topology – The Borel class of a countable union of \$G_delta\$-sets, that are absolute \$F_{sigmadelta}\$ Answer

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gn.common topology – The Borel class of a countable union of \$G_delta\$-sets, that are absolute \$F_{sigmadelta}\$

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Theorem. Each $$G_{deltasigma}$$-subset $$A$$ of a Polish house $$X$$ can breathe written because the union $$bigcup_{ninomega}A_n$$ of a sequence $$(A_n)_{ninomega}$$ of pairwise disjoint $$G_delta$$-sets in $$X$$.

Proof. Write the clique $$A$$ because the union $$A=bigcup_{ninomega}B_n$$ of $$G_delta$$-sets $$B_n$$ in $$X$$ such that $$emptyset=B_0subseteq B_nsubseteq B_{n+1}$$ for all $$n$$. For each $$ninomega$$ write the $$G_delta$$-set $$B_n$$ because the intersection $$B_n=bigcap_{minomega}U_{n,m}$$ of launch units $$U_{n,m}$$ such that $$U_{n,m+1}subseteq U_{n,m}subseteq U_{n,0}=X$$ for all $$m$$. Observe that $${U_{n,m}:n,minomega}subseteq Delta^0_2(X)$$. By Exercise 2.20 in Kechris’ “Classical Descriptive Set Theory”, there exists a stronger Polish topology $$tau’subseteq Sigma^0_2(X)$$ on $$X$$ such every clique $$U_{n,m}$$ is clopen within the topology $$tau’$$. Observe that $$B_{n+1}setminus B_n=bigcup_{minomega}B_{n+1}cap U_{n,m}setminus U_{n,m+1}$$ and every clique $$B_{n+1}cap U_{n,m}setminus U_{n,m+1}$$ is closed within the topology $$tau’$$ and therefore of kind $$G_delta$$ within the topology of the house $$X$$. Now we behold that the $$G_{deltasigma}$$-set $$A$$ is the union $$A=bigcup_{ninomega}B_{n+1}setminus B_n=bigcup_{ninomega}(B_{n+1}cap U_{n,m}setminus U_{n,m+1})$$of the countable household $$massive(B_{n+1}cap U_{n,m}setminus U_{n,m+1}massive)_{n,minomega}$$of pairwise disjoint $$G_delta$$-subsets of $$X$$. $$quadsquare$$

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