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Continuous transformations in R2

  • T. Rado
  • P. V. Reichelderfer
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (GL, volume 75)

Abstract

Throughout Part VI, we shall be concerned with bounded continuous transformations
$$ T:D \to {R^2} $$
(1)
where D is a bounded domain in R2. It will be convenient to use certain alternative representations for T. Note that the points of R 2 are ordered pairs (x, y) of real numbers x, y. Hence we can associate with the point (x, y) of R2 the complex number z = x + iy. In the sequel, the terms “complex number” and “point of R2” will be used interchangeably. Using complex numbers, the transformation (1) can be represented in the form
$$ T:z = T\left( w \right),w \in D $$
(2)
where z is the image point of w under T. Thus T is thought of now as a bounded, continuous, real or complex-valued function of the complex number wD. If we set w = u + iv, z = x + iy, where u, v, x, y are real numbers, then we obtain for T the representation
$$ T:x = x\left( {u,v} \right),y = y\left( {u,v} \right),\left( {u,v} \right) \in D $$
(3)
where the coordinate functions x(u, v), y(u, v) are bounded, continuous, real-valued functions in D.

Keywords

Topological Index Simple Closed Curve Continuous Transformation Oriented Boundary Jordan Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag OHG. in Berlin, Göttingen and Heidelberg 1955

Authors and Affiliations

  • T. Rado
    • 1
  • P. V. Reichelderfer
    • 1
  1. 1.The Ohio State UniversityUSA

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