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
- Uri Abraham, Matatyahu Rubin, and Saharon Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph _1$-dense real order types, Ann. Pure Appl. Logic 29 (1985), no. 2, 123–206. MR 801036, DOI 10.1016/0168-0072(84)90024-1
- James E. Baumgartner, All $\aleph _{1}$-dense sets of reals can be isomorphic, Fund. Math. 79 (1973), no. 2, 101–106. MR 317934, DOI 10.4064/fm-79-2-101-106
- K. F. Barth and W. J. Schneider, Entire functions mapping countable dense subsets of the reals onto each other monotonically, J. London Math. Soc. (2) 2 (1970), 620–626. MR 269834, DOI 10.1112/jlms/2.Part_{4}.620
- Maxim R. Burke and Arnold W. Miller, Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set, Canad. J. Math. 57 (2005), no. 6, 1139–1154. MR 2178555, DOI 10.4153/CJM-2005-044-x
- G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann., 46 (1895) 481–512.
- Paul Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300. MR 98702
- Philip Franklin, Analytic transformations of everywhere dense point sets, Trans. Amer. Math. Soc. 27 (1925), no. 1, 91–100. MR 1501300, DOI 10.1090/S0002-9947-1925-1501300-2
- Lothar Hoischen, Eine Verschärfung eines approximationssatzes von Carleman, J. Approximation Theory 9 (1973), 272–277 (German). MR 367217, DOI 10.1016/0021-9045(73)90093-2
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
- W. D. Maurer, Conformal equivalence of countable dense sets, Proc. Amer. Math. Soc. 18 (1967), 269–270. MR 215994, DOI 10.1090/S0002-9939-1967-0215994-8
- Z. A. Melzak, Existence of certain analytic homeomorphisms, Canad. Math. Bull. 2 (1959), 71–75. MR 105474, DOI 10.4153/CMB-1959-010-8
- J. W. Nienhuys and J. G. F. Thiemann, On the existence of entire functions mapping countable dense sets onto each other, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 4, 331–334. MR 0460638, DOI 10.1016/1385-7258(76)90073-1
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Daihachiro Sato and Stuart Rankin, Entire functions mapping countable dense subsets of the reals onto each other monotonically, Bull. Austral. Math. Soc. 10 (1974), 67–70. MR 346157, DOI 10.1017/S0004972700040636
- Saharon Shelah, Independence results, J. Symbolic Logic 45 (1980), no. 3, 563–573. MR 583374, DOI 10.2307/2273423
- Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
- Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
- Robert J. Zimmer, Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1990. MR 1045444
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