Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$, $ \vert D^if(x)-D^ig(x)\vert<\epsilon(x)$.

References [Enhancements On Off] (What's this?)

  • [ARS] U. Abraham, M. Rubin, S. 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) 123-206. MR 801036 (87d:03132)
  • [Ba] J.E. Baumgartner, All $ \aleph _{1}$-dense sets of reals can be isomorphic, Fund. Math., 79 (1973) 101-106. MR 0317934 (47:6483)
  • [BS] K.F. Barth, 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 0269834 (42:4729)
  • [BM] M.R. Burke, A.W. Miller, Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set, Canad. J. Math., 57 (2005) 1139-1154. MR 2178555 (2006g:03080)
  • [Ca] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann., 46 (1895) 481-512.
  • [Er] P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957) 291-300. MR 0098702 (20:5157)
  • [Fr] P. Franklin, Analytic transformations of everywhere dense point sets, Trans. Amer. Math. Soc., 27 (1925) 91-100. MR 1501300
  • [Ho] L. Hoischen, Eine Verschärfung eines approximationssatzes von Carleman, J. Approximation Theory, 9 (1973) 272-277. MR 0367217 (51:3459)
  • [Je] T. Jech, Set Theory, Academic Press, 1978. MR 506523 (80a:03062)
  • [Ku] K. Kunen, Set Theory, North-Holland, 1983. MR 756630 (85e:03003)
  • [Ma] W.D. Maurer, Conformal equivalence of countable dense sets, Proc. Amer. Math. Soc., 18 (1967) 269-270. MR 0215994 (35:6829)
  • [Me] Z.A. Melzak, Existence of certain analytic homeomorphisms, Canad. Math. Bull., 2 (1959) 71-75. MR 0105474 (21:4215)
  • [NT] J.W. Nienhuys, 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) 331-334. MR 0460638 (57:631)
  • [Ru] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. MR 0924157 (88k:00002)
  • [SR] D. Sato, S. Rankin, Entire functions mapping countable dense subsets of the reals onto each other monotonically, Bull. Austral. Math. Soc., 10 (1974) 67-70. MR 0346157 (49:10883)
  • [Sh1980] S. Shelah, Independence results, J. Symbolic Logic, 45 (1980) 563-573. MR 583374 (82b:03099)
  • [Sh1998] -, Proper and improper forcing, 2nd ed., Springer-Verlag, Berlin, 1998. MR 1623206 (98m:03002)
  • [Wh] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934) 63-89. MR 1501735
  • [Zi] R.J. Zimmer, Essential results of functional analysis, University of Chicago Press, 1990. MR 1045444 (91h:46002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E35, 30E10

Retrieve articles in all journals with MSC (2000): 03E35, 30E10

Additional Information

Maxim R. Burke
Affiliation: Department of Mathematics and Statistics, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3

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

American Mathematical Society