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Effective subsets under homeomorphisms of \(\mathbb{R}^n\). (English) Zbl 1337.03066
This paper studies the (non-)preservation of computability of a compact subset of \(\mathbb{R}^n\) under arbitrary (i.e. not necessarily computable) homeomorphisms.
Strengthening the result that any non-empty computable compact subset of \(\mathbb{R}^n\) is homeomorphic to a non-computable compact subset of \(\mathbb{R}^n\), the paper proves a conjecture of Braverman. Namely, there is a computably (co-)enumerable compact subset K of \([0,1]^n\) such that under no homeomorphism \(f\) on \(\mathbb{R}^n\) do we have that \(f(K)\) is computable.
Reviewer: Danko Ilik (Paris)
MSC:
03D78 Computation over the reals, computable analysis
03F60 Constructive and recursive analysis
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References:
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