Structural properties and Sigma^0_2 enumeration degrees

We prove that each ∑02 set which is hypersimple relative to 0′ is noncuppable in the structure of the ∑02numeration degrees. This gives a connection between properties of ∑02ets under inclusion and and the ∑02 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to 0′.