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A blend of methods of recursion theory and topology: A Π10 tree of shadow points

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Abstract.

This paper is a sequel to our [7]. In that paper we constructed a Π10 tree of avoidable points. Here we construct a Π10 tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2. One interesting fact is that while our Π10 tree of sharp filters (a subtree of Δ) is isomorphic to the tree of addresses (a subtree of 2), the tree of addresses is recursively enumerable but not recursive. To achieve this end we use a finite injury priority argument.

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Correspondence to Iraj Kalantari.

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Mathematics Subject Classification (2000): 03D45, 03D80, 03C57, 54A20

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Kalantari, I., Welch, L. A blend of methods of recursion theory and topology: A Π10 tree of shadow points. Arch. Math. Logic 43, 991–1008 (2004). https://doi.org/10.1007/s00153-004-0244-0

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