No map exists

Prove that there exists no continuous and $1-1$ map ( depiction ) from a sphere to a proper subset of it.

Solution

Let $x_0 \in \mathbb{S}_n$ and consider $f:\mathbb{S}_n \rightarrow \mathbb{S}_n \setminus \{x_0\}$ that is a continous and $1-1$ map . We consider the stereographic projection $g:\mathbb{S}_n \setminus \{x_0\} \rightarrow \mathbb{R}^n$ that is also $1-1$ and continous. Hence the composition $\left( g \circ f \right): \mathbb{S}_n \rightarrow \mathbb{R}^n$ is continuous and $1-1$ which is an obscurity due to Borsuk-Ulam.

The exercise can also be found at mathematica.gr .

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No map exists

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