# How to prove that $R^n\setminus \{0\}$ is not contractible

If I must prove that $$R^{n}\setminus\{0\}$$ is not contractible, how may I do so formally. Using the intuitive notion of contractibility at a point as being that any surface homeomorphic to an $$n$$ sphere, I was thinking that it could be proven by contradiction as follows: Assume for the sake of contradiction that $$R^{n}\setminus\{0\}$$ is contractible. Then, as $$R^{n}\setminus\{0\}$$ can be continuously mapped to $$S^{n-1}$$, $$S^{n-1}$$ must also be contractible. However, as the only shape homeomorphic to $$S^{n-1}$$ that passes through the point under consideration is $$S^{n-1}$$ itself, there exists no contraction at that point of an $$S^{n-1}$$ sphere to a point. Thus, we have reached a contradiction, and $$R^{n}\setminus\{0\}$$ is not contractible. If indeed the formal definition of contractibility (in terms of null homotopy)is equivalent to the one I have used, could you please tell me a source for the same? (so that I may cite it for an assignment)

• you said " $\mathbf{R}^n/ \{0\}$ can be continuously mapped to $S^{n-1}$", Is it $\mathbf{R}^n/ \{0\}$ or $\mathbf{R}^n-\{0\}$? Jan 4, 2019 at 9:58
• When you write "as $R^n \setminus \{0\}$ can be continuously mapped to $S^{n-1}$", you have to write "as $R^n \setminus \{0\}$ is homotopy equivalent to $S^{n-1}$". If you have a continous map from $X$ to $Y$ and $X$ is contractible, $Y$ does not need to be contractible. For example, the inclusion $\{N\} \hookrightarrow S^1$ of the north-pole to the circle is continous and $\{N\}$ is contractible, but $S^1$ is not. Jan 4, 2019 at 9:59
• If it is $\mathbf{R}^n -\{0\}$, homotopy equivalent does the job. If Contractible, $S^{n-1} \sim\mathbf{R}^n -\{0\}\sim *$. Contradiction Jan 4, 2019 at 10:01
• See this question of the user: We are really talking about $R^n \setminus \{0\}$, not $R^n/\{0\}$, which would be $R^n$ anyway. Jan 4, 2019 at 10:03
• I don't understand what you mean with a "shape homeomorphic to $S^{n-1}$ that passes through the point of consideration". Neither do I really understand your "informal" definition of contractability. I think it would also help if you could state the definition of contracibility you know. Jan 4, 2019 at 10:22