TITLE: Homeomorphism between $\mathbb{R}^{n+1} \setminus \{0\}$ and $S^n\times\mathbb{R}$
QUESTION [6 upvotes]: How to establish a homeomorphism between $\mathbb{R}^{n+1} \setminus \{0\}$ and $S^n\times\mathbb{R}$? 
Thanks.

REPLY [7 votes]: Consider the map $f:\mathbb{R}^{n+1}\setminus\{0\} \to S^n \times \mathbb{R}_{++}$ given by $$f(x):=(x/||x||,||x||).$$ It's a homeomorphism, you just need to compose it with a homeomorphism $\mathbb{R}_{++}\to \mathbb{R}$.