Notre Dame Journal of Formal Logic 59 (4):605-636 (2018)

Abstract
We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete metric space cannot be decomposed into countably many nowhere dense pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets.
Keywords Baire category   Weihrauch lattice   genericity   reverse mathematics  computable analysis
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DOI 10.1215/00294527-2018-0016
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References found in this work BETA

Effective Borel Measurability and Reducibility of Functions.Vasco Brattka - 2005 - Mathematical Logic Quarterly 51 (1):19-44.
Closed Choice and a Uniform Low Basis Theorem.Vasco Brattka, Matthew de Brecht & Arno Pauly - 2012 - Annals of Pure and Applied Logic 163 (8):986-1008.

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Completion of Choice.Vasco Brattka & Guido Gherardi - 2021 - Annals of Pure and Applied Logic 172 (3):102914.

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