Topological equivalence to a projection

Abstract

We present a necessary and sufficient condition for a continuous function on a plane to be topologically equivalent to a projection onto one of the coordinates.

Key words: Topological equivalence, projection.

Full Text

Article Information

Title	Topological equivalence to a projection
Source	Methods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 3-5
MathSciNet	3407916
zbMATH	06533463
Milestones	Received 25/11/2014
Copyright	The Author(s) 2015 (CC BY-SA)

Authors Information

V. V. Sharko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

Yu. Yu. Soroka
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine 

Citation Example

V. V. Sharko and Yu. Yu. Soroka, Topological equivalence to a projection, Methods Funct. Anal. Topology 21 (2015), no. 1, 3-5.

BibTex

@article {MFAT762,
    AUTHOR = {Sharko, V. V. and Soroka, Yu. Yu.},
     TITLE = {Topological equivalence to a projection},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {3-5},
      ISSN = {1029-3531},
  MRNUMBER = {3407916},
 ZBLNUMBER = {06533463},
       URL = {http://mfat.imath.kiev.ua/article/?id=762},
}

References

1. V. I. Arnol′d, Topological classification of Morse polynomials, Tr. Mat. Inst. Steklova 268 (2010), no. Differentsialnye Uravneniya i Topologiya. I, 40-55.  MathSciNet  CrossRef
2. William M. Boothby, The topology of the level curves of harmonic functions with critical points, Amer. J. Math. 73 (1951), 512-538.  MathSciNet 
3. William M. Boothby, The topology of regular curve families with multiple saddle points, Amer. J. Math. 73 (1951), 405-438.  MathSciNet 
4. Wilfred Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154-185.  MathSciNet 
5. James A. Jenkins, Marston Morse, Contour equivalent pseudoharmonic functions and pseudoconjugates, Amer. J. Math. 74 (1952), 23-51.  MathSciNet 
6. James A. Jenkins, Marston Morse, Topological methods on Riemann surfaces. Pseudoharmonic functions, in: Contributions to the theory of Riemann surfaces, Princeton University Press, Princeton, N. J., 1953.  MathSciNet 
7. James Jenkins, Marston Morse, Conjugate nets on an open Riemann surface, in: Lectures on functions of a complex variable, The University of Michigan Press, Ann Arbor, 1955.  MathSciNet 
8. Marston Morse, The topology of pseudo-harmonic functions, Duke Math. J. 13 (1946), 21-42.  MathSciNet 
9. Marston Morse, Topological Methods in the Theory of Functions of a Complex Variable, Princeton University Press, Princeton, N. J., 1947.  MathSciNet 
10. A. A. Oshemkov, Morse functions on two-dimensional surfaces. Coding of singularities, Trudy Mat. Inst. Steklov. 205 (1994), no. Novye Rezult. v Teor. Topol. Klassif. Integr. Sistem, 131-140.  MathSciNet 
11. E. Polulyakh, I. Yurchuk, On the pseudo-harmonic functions defined on a disk, Proceedings of the Institute of Mathematics of NAS of Ukraine, Kyiv, vol. 8, 2009.  MathSciNet 
12. V. V. Sharko, Smooth and topological equivalence of functions on surfaces, Ukrain. Mat. Zh. 55 (2003), no. 5, 687-700.  MathSciNet  CrossRef
13. V. V. Sharko, Topological equivalence of harmonic polynomials, Zb. prac Inst. mat. NAN Ukr., Kyiv 10 (2013), no. 4-5, 542-551. (Russian)