Logic: from foundations to applications
Proof Mining in R-trees and Hyperbolic Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
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We extend the principle @?"1^0-UB of uniform @?"1^0-boundedness introduced earlier by the author to a uniform boundedness principle @?-UB^X for abstract bounded metric and hyperbolic spaces which are not assumed to be compact. Despite the fact that this principle implies numerous results which in general are true only for compact spaces (and continuous functions) we can prove that for a large class K of such consequences A the conclusion A is true in arbitrary bounded spaces even when @?-UB^X is used to facilitate the proof of A. For a somewhat more restricted class of sentences A even effective uniform bounds can be extracted from such proofs.