P01\it \Pi^0_1 Sets and Tilings

In this paper, we prove that given any \(\it \Pi^0_1\) subset P of {0,1}ℕ there is a tileset τ with a countable set of configurations C such that P is recursively homeomorphic to C ∖ U where U is a computable set of configurations. As a consequence, if P is countable, this tileset has the exact same set of Turing degrees.