We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire space.

In this article we investigate the dual-shattering cardinal H, the dualsplitting cardinal S and the dual-reaping cardinal R, which are dualizations of the well-known cardinals h (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and r (the reaping number). Using some properties of the ideal J of nowhere dual-Ramsey sets, which is an ideal ov...