Homotopy Groups

Abstract

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.