Entire functions mapping uncountable dense sets of reals onto each other monotonically
Author:
Maxim R. Burke
Journal:
Trans. Amer. Math. Soc. 361 (2009), 28712911
MSC (2000):
Primary 03E35; Secondary 30E10
Published electronically:
January 22, 2009
MathSciNet review:
2485411
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Additional Information
Abstract: When and are countable dense subsets of , it is a wellknown result of Cantor that and are orderisomorphic. A theorem of K.F. Barth and W.J. Schneider states that the orderisomorphism can be taken to be very smooth, in fact the restriction to of an entire function. J.E. Baumgartner showed that consistently and any two subsets of having points in every interval are orderisomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the orderisomorphism cannot be taken to be smooth. A useful variant of Baumgartner's result for second category sets was established by S. Shelah. He showed that it is consistent that and second category sets of cardinality exist while any two sets of cardinality which have second category intersection with every interval are orderisomorphic. In this paper, we show that the orderisomorphism in Shelah's theorem can be taken to be the restriction to of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer , a nondecreasing surjection of class and a positive continuous function , we may choose the orderisomorphism so that for all and for all , .
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Additional Information
Maxim R. Burke
Affiliation:
Department of Mathematics and Statistics, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3
Email:
burke@upei.ca
DOI:
http://dx.doi.org/10.1090/S0002994709049241
PII:
S 00029947(09)049241
Keywords:
Orderisomorphism,
second category,
entire function,
oraclecc forcing,
Carleman's theorem,
Hoischen's theorem
Received by editor(s):
March 10, 2006
Published electronically:
January 22, 2009
Additional Notes:
The author’s research was supported by NSERC. The author thanks F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when much of the present paper was written.
Article copyright:
© Copyright 2009
American Mathematical Society
