A note on an effective Polish topology and Silver's Dichotomy theorem

TL;DR

This paper introduces a new Polish topology inspired by the Gandy-Harrington topology, which facilitates a proof of Silver's dichotomy theorem within the Polish setting, highlighting a decomposition of certain equivalence relations.

Contribution

It defines a novel Polish topology that simplifies the proof of Silver's dichotomy theorem and characterizes the decomposition of $ ext{Pi}^1_1$ equivalence relations.

Findings

Decomposition of $ ext{Pi}^1_1$ relations into clopen and meager parts

Largest regular topology with a basis in $ ext{Sigma}^1_1$

Topology inspired by Gandy-Harrington used to prove Silver's dichotomy

Abstract

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. In this topology, a $\Pi^1_1$ equivalence relation decomposes into a "sum" of a clopen relation and a meager one. We characterize it as the largest regular toplogy with a basis included in $\Sigma^1_1$.