Homotopy Groups

  • Gregory L. Naber
Part of the Texts in Applied Mathematics book series (TAM, volume 25)


The real line ℝ is not homeomorphic to the plane ℝ2, but this fact is not quite the triviality one might hope. Perhaps the most elementary proof goes as follows: Suppose there were a homeomorphism h of ℝ onto ℝ2. Select some point x 0 ∈ ℝ. The restriction of h to ℝ − {x 0} would then carry it homeomorphically onto ℝ2 − {h(x 0)}. However, \(\mathbb{R} -\{ {x}_{0}\} = (-\infty,{x}_{0}) \cup({x}_{0},\infty )\) is not connected, whereas ℝ2 − {h(x 0)} certainly is connected (indeed, pathwise connected). Since connectedness is a topological property, this cannot be and we have our contradiction.


Topological Space Base Point Fundamental Group Homotopy Class Homotopy Group 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Gregory L. Naber
    • 1
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA

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