A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

We extend the principle -UB of uniform -boundedness introduced earlier by the author to a uniform boundedness principle ∃-UBX 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 ∃−UBX 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.