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

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- by Maxim R. Burke PDF
- Trans. Amer. Math. Soc.
**361**(2009), 2871-2911 Request permission

## Abstract:

When $A$ and $B$ are countable dense subsets of $\mathbb {R}$, it is a well-known result of Cantor that $A$ and $B$ 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 $\mathbb {R}$ of an entire function. J.E. Baumgartner showed that consistently $2^{\aleph _0}>\aleph _1$ and any two subsets of $\mathbb {R}$ having $\aleph _1$ 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 $2^{\aleph _0}>\aleph _1$ and second category sets of cardinality $\aleph _1$ exist while any two sets of cardinality $\aleph _1$ 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 $\mathbb {R}$ of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer $n$, a nondecreasing surjection $g\colon \mathbb {R}\to \mathbb {R}$ of class $C^n$ and a positive continuous function $\epsilon \colon \mathbb {R}\to \mathbb {R}$, we may choose the order-isomorphism $f$ so that for all $i=0,1,\dots ,n$ and for all $x\in \mathbb {R}$, $|D^if(x)-D^ig(x)|<\epsilon (x)$.## References

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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
- 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.
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**361**(2009), 2871-2911 - MSC (2000): Primary 03E35; Secondary 30E10
- DOI: https://doi.org/10.1090/S0002-9947-09-04924-1
- MathSciNet review: 2485411