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Every co-c.e. closed set (Π0 1 class) in Cantor space is represented by a co-c.e. tree. Our aim is to clarify the interaction between the Medvedev and Muchnik degrees of co-c.e. closed subsets of Cantor space and the Turing degrees of their co-c.e. representations. Among other results, we present the following theorems: if v and w are different c.e. degrees, then the collection of the Medvedev (Muchnik) degrees of all Π0 1 classes represented by v and the collection represented by w are also different; the ideals generated from such collections are also different; the collections of the Medvedev and Muchnik degrees of all Π0 1 classes represented by incomplete co-c.e. sets are upward dense; the collection of all Π0 1 classes represented by K-trivial sets is Medvedev-bounded by a single Π0 1 class represented by an incomplete co-c.e. set; and the Π0 1 classes have neither nontrivial infinite suprema nor infima in the Medvedev lattice.
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