Forcing $\hat{\mathbf{I}}{\pounds }_1^1$-Separation on $\ddot{I}{‰}_1^{\ddot{I}{‰}_1}$

Abstract

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