Topology, Geometry and Gauge fields pp 97-156 | Cite as

# Homotopy Groups

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## Abstract

The real line ℝ is *not* homeomorphic to the plane ℝ^{2}, 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 ℝ^{2}. Select some point *x* _{0} ∈ ℝ. The restriction of *h* to ℝ − {*x* _{0}} would then carry it homeomorphically onto ℝ^{2} − {*h*(*x* _{0})}. However, \(\mathbb{R} -\{ {x}_{0}\} = (-\infty,{x}_{0}) \cup({x}_{0},\infty )\) is not connected, whereas ℝ^{2} − {*h*(*x* _{0})} certainly is connected (indeed, pathwise connected). Since connectedness is a topological property, this cannot be and we have our contradiction.

## Keywords

Topological Space Base Point Fundamental Group Homotopy Class Homotopy Group
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.

## References

- Mass.Massey, W.S.,
*Algebraic Topology: An Introduction*, Springer-Verlag, New York, Berlin, 1991.Google Scholar - Gra.Gray, B.,
*Homotopy Theory: An Introduction to Algebraic Topology*, Academic Press, New York, 1975.Google Scholar - Rav.Ravenel, D.C.,
*Complex Cobordism and Stable Homotopy Groups of Spheres*, Academic Press, Orlando, 1986.Google Scholar - Dug.Dugundji, J.,
*Topology*, Allyn and Bacon, Boston, 1966.Google Scholar

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