A note on an effective Polish topology and Silver’s dichotomy theorem

Sami, Ramez

doi:10.1090/proc/14541

A note on an effective Polish topology and Silver’s dichotomy theorem
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by Ramez L. Sami PDF

Proc. Amer. Math. Soc. 147 (2019), 4039-4044 Request permission

Abstract:

$\boldsymbol {\cdot }$ We define a Polish topology inspired from the Gandy-Harrington topology and show how it can be used to prove Silver’s dichotomy theorem while remaining in the Polish realm.

$\boldsymbol {\cdot }$ In this topology, a $\Pi _{1}^{1}$ equivalence decomposes into a “sum” of a clopen relation and a meager one.

$\boldsymbol {\cdot }$ We characterize it as the largest regular topology with a basis included in $\Sigma _{1}^{1}$.

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