On monotone presentations of Borel sets. (English) Zbl 1187.03038

Let \((X,\tau)\) be an uncountable Polish space and let \(\xi\geq2\) be a countable ordinal number. The authors prove that there are no functions \(f_n:\boldsymbol\Sigma^0_\xi\to\boldsymbol\Pi^0_\xi\) for \(n\in\omega\) such that for all \(Q,Q'\in\boldsymbol\Sigma^0_\xi\), \(Q\subseteq Q'\) implies \(f_n(A)\subseteq f_n(Q')\) and \(Q=\bigcup_{n<\omega}f_n(Q)\). This result answers a question of Márton Elekes in the negative. In fact the authors prove a more general theorem which is also used to prove the nonexistence of a monotone representation for Borel functions.


03E15 Descriptive set theory
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