## Abstract

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 language | English |
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Pages (from-to) | 3367-3377 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 132 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2004 |

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics

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