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SUMMARY:Vittorio Cipriani (University of Udine)
DTSTART:20211011T203000Z
DTEND:20211011T210000Z
DTSTAMP:20260423T004821Z
UID:CTA/62
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CTA/62/">Can
 tor-Bendixson Theorem in the Weihrauch lattice</a>\nby Vittorio Cipriani (
 University of Udine) as part of Computability theory and applications\n\n\
 nAbstract\nIn this talk we will study the Cantor-Bendixson theorem using t
 he\nframework of Weihrauch reducibility. (Variations of) this theorem fall
 s into\nthe highest of the big-five axiom systems of reverse mathematics\,
  namely\n\\Pi_1^1-CA_0 . Kihara\, Marcone and Pauly already showed that ma
 ny problems\nrepresenting principles equivalent to ATR_0 lie in different 
 Weihrauch\ndegrees\, for \\Pi_1^1-CA_0 we actually have a natural candidat
 e\, namely the one\nmapping a countable sequence of trees to the character
 istic function of the\nset of indices corresponding to well-founded trees.
  This principle was firstly\nconsidered by Hirst\, that also showed its We
 ihrauch equivalence with PK_Tr\,\nthe function that takes as input a tree 
 and outputs its perfect kernel for\ntrees. Even if in reverse mathematics 
 it is actually equivalent to consider\ntrees or closed sets\, we will show
  that PK_Tr <_W PK_X\, where PK_X takes as\ninput a closed set of a comput
 able Polish space X and outputs its perfect\nkernel. The equivalence betwe
 en these two shows up if we switch to\narithmetical Weihrauch reducibility
 .\n\nWe will continue in this direction showing (non) reductions between p
 roblems\nrelated to the Cantor-Bendixson theorem with particular attention
  paid to\nclassify them for every computable Polish space X. This leads us
  to the result\nthat\, while PK_X and wCB_X (i.e. same as PK_X but where t
 he output also\nprovides a listing of the elements in the scattered part) 
 are equivalent for\nany space X that we consider\, the problem CB_X (i.e. 
 same as wCB_X but where\nthe output provides also the cardinality of the s
 cattered part) "almost"\nsplits in two Weihrauch degrees\, one having as r
 epresentative PK_X and the\nother having CB_\\Baire.\n\nThis is joint work
  with Alberto Marcone and Manlio Valenti.\n
LOCATION:https://researchseminars.org/talk/CTA/62/
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