23 research outputs found

    The descriptive theory of represented spaces

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    This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

    First Order Theories of Some Lattices of Open Sets

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    We show that the first order theory of the lattice of open sets in some natural topological spaces is mm-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., Rn\mathbb{R}^n, n1n\geq1, and the domain PωP\omega) this theory is mm-equivalent to first order arithmetic

    A generalization of a theorem of Hurewicz for quasi-Polish spaces

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    We identify four countable topological spaces S2S_2, S1S_1, SDS_D, and S0S_0 which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the T2T_2, T1T_1, TDT_D, and T0T_0-separation axioms. S2S_2 is the space of rationals, S1S_1 is the natural numbers with the cofinite topology, SDS_D is an infinite chain without a top element, and S0S_0 is the set of finite sequences of natural numbers with the lower topology induced by the prefix ordering. Our main result is a generalization of Hurewicz's theorem showing that a co-analytic subset of a quasi-Polish space is either quasi-Polish or else contains a countable Π20\Pi^0_2-subset homeomorphic to one of these four spaces

    Levels of discontinuity, limit-computability, and jump operators

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    We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of Δ20\Delta^0_2-measurable functions between arbitrary countably based T0T_0-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem
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