Abstract
We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.
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The research of M. Hrušák partially supported by GA ČR grant 201-03-0933, PAPIIT grant IN106705 and CONACYT grant 46337-F. The research of J. Zapletal partially supported by GA ČR grant 201-03-0933 and NSF grant DMS 0300201. The results contained in the paper were obtained while the second author visited UNAM, Morelia, Mexico.