Computable analysis: an introduction
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Pi-0-1 classes in computable analysis and topology
Pi-0-1 classes in computable analysis and topology
Computability of the Radon-Nikodym derivative
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
Multi-valued functions in computability theory
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. Another main result is that connected choice is complete for dimension greater or equal to three in the sense that it is computably equivalent to Weak Kőnig's Lemma. In contrast to this, the connected choice operations in dimensions zero, one and two form a strictly increasing sequence of Weihrauch degrees, where connected choice of dimension one is known to be equivalent to the Intermediate Value Theorem. Whether connected choice of dimension two is strictly below connected choice of dimension three or equivalent to it is unknown, but we conjecture that the reduction is strict. As a side result we also prove that finding a connectedness component in a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig's Lemma.