Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Covering $\mathbb R^{n+1}$ by graphs of $n$-ary functions and long linear orderings of Turing degrees

Authors: Uri Abraham and Stefan Geschke
Journal: Proc. Amer. Math. Soc. 132 (2004), 3367-3377
MSC (2000): Primary 03E17, 03E25; Secondary 26A99, 26B99
Published electronically: June 18, 2004
MathSciNet review: 2073314
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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.

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Additional Information

Uri Abraham
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

Stefan Geschke
Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin

Keywords: Continuous function, $n$-space, forcing extension, covering number, Turing degree
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.
Article copyright: © Copyright 2004 American Mathematical Society