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Isomorphism of locally compact Polish metric structures (2209.05903v1)
Published 13 Sep 2022 in math.LO
Abstract: We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\pmb{\Pi}0_{\alpha+1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha$, $\alpha \geq 2$. We also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures.