Entire functions mapping uncountable dense sets of reals onto each other monotonically

Author:
Maxim R. Burke

Journal:
Trans. Amer. Math. Soc. **361** (2009), 2871-2911

MSC (2000):
Primary 03E35; Secondary 30E10

Published electronically:
January 22, 2009

MathSciNet review:
2485411

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Abstract: When and are countable dense subsets of , it is a well-known result of Cantor that and are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism 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 order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism 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 order-isomorphic. In this paper, we show that the order-isomorphism 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 order-isomorphism 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/S0002-9947-09-04924-1

Keywords:
Order-isomorphism,
second category,
entire function,
oracle-cc 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