Covering $\mathbb R^{n+1}$ by graphs of $n$-ary functions and long linear orderings of Turing degrees
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- by Uri Abraham and Stefan Geschke PDF
- Proc. Amer. Math. Soc. 132 (2004), 3367-3377 Request permission
Abstract:
A point $(x_0,\dots ,x_n)\in X^{n+1}$ is covered by a function $f:X^n\to X$ iff there is a permutation $\sigma$ of $n+1$ such that $x_{\sigma (0)}=f(x_{\sigma (1)},\dots ,x_{\sigma (n)})$. By a theorem of Kuratowski, for every infinite cardinal $\kappa$ exactly $\kappa$ $n$-ary functions are needed to cover all of $(\kappa ^{+n})^{n+1}$. We show that for arbitrarily large uncountable $\kappa$ it is consistent that the size of the continuum is $\kappa ^{+n}$ and $\mathbb R^{n+1}$ is covered by $\kappa$ $n$-ary continuous functions. We study other cardinal invariants of the $\sigma$-ideal on $\mathbb R^{n+1}$ generated by continuous $n$-ary functions and finally relate the question of how many continuous functions are necessary to cover $\mathbb R^2$ to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.References
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Additional Information
- Uri Abraham
- Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel
- Email: abraham@math.bgu.ac.il
- Stefan Geschke
- Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin
- MR Author ID: 681801
- Email: geschke@math.fu-berlin.de
- Received by editor(s): November 12, 2002
- Received by editor(s) in revised form: July 22, 2003
- Published electronically: June 18, 2004
- Communicated by: Carl G. Jockusch, Jr.
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3367-3377
- MSC (2000): Primary 03E17, 03E25; Secondary 26A99, 26B99
- DOI: https://doi.org/10.1090/S0002-9939-04-07422-2
- MathSciNet review: 2073314