Covering ℝ n+1 by graphs of n-ary functions and long linear orderings of turing degrees

Uri Abraham, Stefan Geschke

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


A point (x 0, . . . , x n) ∈ X n+1 is covered by a function f X n → X iff there is a permutation σ of n + 1 such that x σ(0) = f (x σ(1), . . . , x σ(n)). By a theorem of Kuratowski, for every infinite cardinal κ exactly κ n-ary functions are needed to cover all of (κ +n) n+1. We show that for arbitrarily large uncountable κ it is consistent that the size of the continuum is κ +n and ℝ n+1 is covered by κ n-ary continuous functions. We study other cardinal invariants of the σ-ideal on ℝ n+1 generated by continuous n-ary functions and finally relate the question of how many continuous functions are necessary to cover ℝ 2 to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered. Continuous function, n-space, forcing extension, covering number, Turing degree.

Original languageEnglish
Pages (from-to)3367-3377
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number11
StatePublished - 1 Nov 2004

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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