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Located Sets And Reverse Mathematics
 Journal of Symbolic Logic
, 1999
"... Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly loca ..."
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Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets.
Π 0 1 Classes and pseudojump operators
, 2008
"... For a pseudojump V X and a Π 0 1 class P, we consider properties of the set {V X: X ∈ P}. We show that if P is Medvedev complete or if P has positive measure, and ∅ ′ ≤T C, then there exists X ∈ P with V X ≡T C. We examine the consequences when V X is Turing incomparable with V Y for X = Y in P an ..."
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For a pseudojump V X and a Π 0 1 class P, we consider properties of the set {V X: X ∈ P}. We show that if P is Medvedev complete or if P has positive measure, and ∅ ′ ≤T C, then there exists X ∈ P with V X ≡T C. We examine the consequences when V X is Turing incomparable with V Y for X = Y in P and when W X e = W Y e for all X, Y ∈ P. Finally, we give a characterization of the jump in terms of Π 0 1 classes.
Immunity for Closed Sets ⋆
"... Abstract. The notion of immune sets is extended to closed sets and Π 0 1 classes in particular. We construct a Π 0 1 class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A, B) of separating sets for A and B is immune. We show that ..."
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Abstract. The notion of immune sets is extended to closed sets and Π 0 1 classes in particular. We construct a Π 0 1 class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A, B) of separating sets for A and B is immune. We show that every perfect thin Π 0 1 class is immune. We define the stronger notion of prompt immunity and construct an example of a Π 0 1 class of positive measure which is promptly immune. We show that the immune degrees in the Medvedev lattice of closed sets forms a filter. We show that for any Π 0 1 class P with no computable element, there is a Π 0 1 class Q which is not immune and has no computable element, and which is Medvedev reducible to P. We show that any random closed set is immune.