On the computational content of the Baire Category Theorem
Seminar Room 1, Newton Institute
AbstractWe present results on the classification of the computational content of the Baire Category Theorem in the Weihrauch lattice. The Baire Category Theorem can be seen as a pigeonhole principle that states that a large (= complete) metric space cannot be decomposed into a countable number of small (= nowhere dense) pieces (= closed sets). The difficulty of the corresponding computational task depends on the logical form of the statement as well as on the information that is provided. In the first logical form the task is to find a point in the space that is left out by a given decomposition of the space that consists of small pieces. In the contrapositive form the task is to find a piece that is not small in a decomposition that exhausts the entire space. In both cases pieces can be given by descriptions in negative or positive form. We present a complete classification of the complexity of the Baire Category Theorem in all four cases and for certain types of spaces. The results are based on joint work with Guido Gherardi and Alberto Marcone, on the one hand, and Matthew Hendtlass and Alexander Kreuzer, on the other hand. One obtains a refinement of what is known in reverse mathematics in this way.
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volume 2136 of Lecture Notes in Computer Science, pages 224–235, Berlin, 2001.
26th International Symposium, MFCS 2001, Mariánské Lázně, Czech Republic, August 27-31, 2001.
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