zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] Brattka, V., Plottable real number functions and the computable graph theorem, SIAM J. Comput., 38, 1, 303-328, (2008) · Zbl 1165.03052 [2] Rogers, H., Theory of recursive functions and effective computability, (1987), MIT Press Cambridge · Zbl 0183.01401 [3] Weihrauch, K., Computable analysis: an introduction, (2000), Springer Berlin · Zbl 0956.68056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.