The shrinking property for NP and coNP

We study the shrinking and separation properties (two notions well-known in descriptive set theory) for and and show that under reasonable complexity-theoretic assumptions, both properties do not hold for and the shrinking property does not hold for . In particular we obtain the following results. 1. and do not have the shrinking property unless is finite. In general, and do not have the shrinking property unless is finite. This solves an open question posed by Selivanov (1994) [33].2.The separation property does not hold for unless .3.The shrinking property does not hold for unless there exist -hard disjoint -pairs (existence of such pairs would contradict a conjecture of Even et al. (1984) [6]).4.The shrinking property does not hold for unless there exist complete disjoint -pairs. Moreover, we prove that the assumption is too weak to refute the shrinking property for in a relativizable way. For this we construct an oracle relative to which , , and has the shrinking property. This solves an open question posed by Blass and Gurevich (1984) [3] who explicitly ask for such an oracle.