Jind\v rich Zapletal
Forcing with ideals generated by closed sets

Comment.Math.Univ.Carolinae 43,1 (2002) 181-188.

Abstract:Consider the poset $P_I=\text {Borel}(\Bbb R)\setminus I$ where $I$ is an arbitrary $\sigma $-ideal $\sigma $-generated by a projective collection of closed sets. Then the $P_I$ extension is given by a single real $r$ of an almost minimal degree: every real $s\in V[r]$ is Cohen-generic over $V$ or $V[s]=V[r]$.

Keywords: forcing, descriptive set theory, large cardinals
AMS Subject Classification: 03E17, 03E55, 03E60