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We develop a flexible method for showing that Borel witnesses to some combinatorial property of $\Delta ^1_1$ objects yield $\Delta ^1_1$ witnesses. We use a modification the Gandy–Harrington forcing method of proving dichotomies, and we can recover the complexity consequences of many known dichotomies with short and simple proofs. Using our methods, we give a simplified proof that smooth $\Delta ^1_1$ equivalence relations are $\Delta ^1_1$-reducible to equality; we prove effective versions of the Lusin–Novikov and Feldman–Moore theorems; we prove new effectivization results related to dichotomy theorems due to Hjorth and Miller (originally proven using “forceless, ineffective, and powerless” methods); and we prove a new upper bound on the complexity of the set of Schreier graphs for $\mathbb {Z}^2$ actions. We also prove an equivariant version of the $G_0$ dichotomy that implies some of these new results and a dichotomy for graphs induced by Borel actions of $\mathbb {Z}^2$.
It is proved that every remainder of a nonlocally compact semitopological group $G$ is a Baire space if and only if $G$ is not Čech-complete, which improves a dichotomy theorem of topological groups by Arhangel’skiǐ [‘The Baire property in remainders of topological groups and other results’, Comment. Math. Univ. Carolin.50(2) (2009), 273–279], and also gives a positive answer to a question of Lin and Lin [‘About remainders in compactifications of paratopological groups’, ArXiv: 1106.3836v1 [Math. GN] 20 June 2011]. We also show that for a nonlocally compact rectifiable space $G$ every remainder of $G$ is either Baire, or meagre and Lindelöf.
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