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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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