Geodesic
Forcing with ideals generated by closed sets
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 181-188

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Consider the poset $P_I=\text{\rm Borel}(\Bbb R)\setminus I$ 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 $V[s]=V[r]$.
Classification : 03E15, 03E17, 03E40, 03E55, 03E60
Keywords: forcing; descriptive set theory; large cardinals 
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Zapletal, Jindřich. Forcing with ideals generated by closed sets. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 181-188. http://geodesic.mathdoc.fr/item/CMUC_2002__43_1_a16/