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An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a non-measure 0 subset of the set of decidable sequences are Turing reducible to x.

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Abstract. For each pointclass Γ ⊆ P (2 ω ) define U [Γ] as the collection of all X ⊆ 2 ω such that the preimage f −1 (X) belongs to Γ for each continuous f : 2 ω → 2 ω . We study the properties of and possible relationships among the classes U [Γ], where Γ ranges over the σ-algebras (l), (m), the completely Ramsey sets, and the sets with the Baire property. We also prove some results about cardinal coefficients of U [Γ] for the general case of Marczewski-Burstin representable σ-algebras Γ. We finish by posing some unsolved problems.

We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above. MSC:03E15, 03E20, 54A10