2012/12/12 by Luca Motto Ros, Ros, Luca Motto
Mathematics · #03D55 #03D78 #03E15 #03E60 #54C10 #54E40 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03D55 #msc:03D78 #msc:03E15 #msc:03E60 #msc:54C10 #msc:54E40
paper · pdf · doi:10.48550/arxiv.1212.3005
31 pages, 3 figures. Revised version, accepted for publication on Fundamenta Mathematicae
arxiv created 2013/09/12 · arxiv updated 2013/09/13
We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on X (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of X, and we show that the resulting hierarchies of degrees are much more complicated than the classical Wadge hierarchy; in particular, they always contain large infinite antichains, and in most cases also infinite descending chains.