  With every $\sigma$-ideal $I$ on a Polish space we associate the
$\sigma$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing
notions of Borel sets modulo the respective $\sigma$-ideals $I$ and $I^*$ and
find connections between their forcing properties. To this end, we associate to
a $\sigma$-ideal on a Polish space an ideal on a countable set and show how
forcing properties of the forcing depend on combinatorial properties of the
ideal. For $\sigma$-ideals generated by closed sets we also study the degrees
of reals added in the forcing extensions. Among corollaries of our results, we
get necessary and sufficient conditions for a $\sigma$-ideal $I$ generated by
closed sets, under which every Borel function can be restricted to an
$I$-positive Borel set on which it is either 1-1 or constant.
