LOCAL-DENSITY OF KLEENE DEGREES

Concerning Post's problem for Kleene degrees and its relativization, Hrbacek showed in [1] and [2] that if V = L, then Kleene degrees of coanalytic sets are dense, and then for all K subset of or equal to (omega)omega, there are aleph(1) sets which are Kleene semirecursive in K and not Kleene recursive in each other and K. But the density of Kleene semirecursive in K Kleene degrees is not obtained from these theorems. In this note, we extend these theorems by showing that if V = L, then for all K subset of or equal to (omega)omega, Kleene semirecursive in K Kleene degrees are dense above K.