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We show that if F is any " well-behaved " subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on P(ω ω) induced by F turns out to look like the Wadge hierarchy (which is the special case where F is the set of continuous functions).

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called ∆ 0 α-reductions, and try to find for various natural… (More)

We present a general way of defining various reduction games on ω which " represent " corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ-measurable functions. These games turn out to… (More)

In [8] we have considered a wide class of " well-behaved " reducibil-ities for sets of reals. In this paper we continue with the study of Borel re-ducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [8] allowing to deal with a larger class of notions of… (More)

We show that if κ is a weakly compact cardinal then the embed-dability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order on the generalized Cantor space κ 2 there is an L κ + κ-sentence ϕ such that the embeddability relation on its models of size κ, which are all trees, is Borel bireducible… (More)

We analyze the reducibilities induced by, respectively, uniformly continuous, Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces, and determine whether under suitable set-theoretical assumptions the induced degree-structures are well-behaved.

We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σ 0 α-measurable functions (for every fixed 1 ď α ă ω 1). Moreover , we present some results concerning those Borel functions… (More)

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