Covering by graphs of -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A point is *covered* by a function iff there is a permutation of such that .

By a theorem of Kuratowski, for every infinite cardinal exactly -ary functions are needed to cover all of . We show that for arbitrarily large uncountable it is consistent that the size of the continuum is and is covered by -ary continuous functions.

We study other cardinal invariants of the -ideal on generated by continuous -ary functions and finally relate the question of how many continuous functions are necessary to cover 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

Email:
abraham@math.bgu.ac.il

**Stefan Geschke**

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

Email:
geschke@math.fu-berlin.de

DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07422-2

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