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)

 
 

 

Open mappings of the universal curve onto continuous curves


Author: David C. Wilson
Journal: Trans. Amer. Math. Soc. 168 (1972), 497-515
MSC: Primary 54F50
DOI: https://doi.org/10.1090/S0002-9947-1972-0298630-0
MathSciNet review: 0298630
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A criterion for the existence of an open mapping from one compact metric space onto another is established in this paper. This criterion is then used to establish the existence of a monotone open mapping of the universal curve onto any continuous curve and the existence of a light open mapping of the universal curve onto any nondegenerate continuous curve. These examples show that if $ f$ is a monotone open or a light open mapping of one compact space $ X$ onto another $ Y$, then it will not necessarily be the case that $ \dim Y \leqq \dim X + k$, where $ k$ is some positive integer.


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

  • [1] P. Alexandroff, C. R. Acad. URRS 4 (1936), 293.
  • [2] R. D. Anderson, A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. (2) 67 (1958), 313-324. MR 20 #2675. MR 0096180 (20:2675)
  • [3] -, Continuous collections of continuous curves, Duke Math. J. 21 (1954), 363-367. MR 15, 977. MR 0062429 (15:977h)
  • [4] -, Continuous collections of continuous curves in the plane, Proc. Amer. Math. Soc. 3 (1952), 647-657. MR 14, 783. MR 0053497 (14:783c)
  • [5] -, A continuous curve admitting monotone open maps onto all locally connected metric continua, Bull. Amer. Math. Soc. 62 (1956), 264-265.
  • [6] -, Monotone interior dimension-raising mappings, Duke Math. J. 19 (1952), 359-366. MR 14, 71. MR 0048798 (14:71c)
  • [7] -, One-dimensional continuous curves, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 760-762. MR 18, 325. MR 0080908 (18:325d)
  • [8] -, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math.(2), 68 (1958), 1-16. MR 20 #2676. MR 0096181 (20:2676)
  • [9] -, Open mappings of compact continua, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 347-349. MR 17, 1230. MR 0078682 (17:1230h)
  • [10] -, Open mappings of continua, Summer Institute on Set Theoretic Topology, Amer. Math. Soc., Providence, R. I., 1958.
  • [11] R.H. Bing, Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101-1110. MR 11, 733. MR 0035429 (11:733i)
  • [12] -, Partitioning continuous curves, Bull. Amer. Math. Soc. 58 (1952), 536-556. MR 14, 192. MR 0049550 (14:192e)
  • [13] E. Dyer, Certain transformations which lower dimension, Ann. of Math. (2) 63 (1956), 15-19. MR 17, 993. MR 0077117 (17:993a)
  • [14] E. Dyer and M. E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103-118. MR 19, 1187. MR 0092959 (19:1187e)
  • [15] L. Keldyš, Example of a one-dimensional continuum with a zero-dimensional and interior mapping onto the square, Dokl. Akad. Nauk SSSR 97 (1954), 201-204. (Russian) MR 16, 60. MR 0063030 (16:60f)
  • [16] A. Kolmogoroff, Über offene Abbildungen, Ann. of Math. (2) 38 (1937), 36-38.
  • [17] L. F. McAuley, Open mappings and open problems, Topology Conference (Arizona State University, 1967), Arizona State Univ., Tempe, Ariz., 1968, pp. 184-202. MR 39 #2134. MR 0240789 (39:2134)
  • [18] G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I., 1963. MR 32 #425. MR 0182943 (32:425)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54F50

Retrieve articles in all journals with MSC: 54F50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0298630-0
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society