## 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

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