Forcing $\mathbf{\Sigma}^1_1$-Separation on $\omega_1^{\omega_1}$

TL;DR

This paper demonstrates the consistency of separating disjoint analytic subsets of a certain uncountable function space using a definable set, employing a forcing method starting from the constructible universe.

Contribution

It introduces a forcing construction that ensures $oldsymbol{ ext{Sigma}}^1_1$-separation on $oldsymbol{ extomega}_1^{oldsymbol{ extomega}_1}$ while preserving CH and the structure of $oldsymbol{ extomega}_1$.

Findings

Disjoint $oldsymbol{ extSigma}^1_1$ sets can be separated by $oldsymbol{ extDelta}^1_1$ sets in the constructed model.

The forcing preserves CH and the cardinal structure of $oldsymbol{ extomega}_1$.

The method starts from the constructible universe $L$.

Abstract

We prove that it is consistent that every two disjoint boldface $\mathbf{\Sigma}^1_1$ subsets of $\omega_1^{\omega_1}$ can be separated by a boldface $\mathbf{\Delta}^1_1$ set. The forcing starts from $L$ and preserves CH and therefore also $\omega_1^{<\omega_1}=\omega_1$.