Could someone please tell me why is $\displaystyle S^n-{x}$ isomorphic to $\displaystyle \mathbb{R}^n$?
Thank you so much.
Well, that depends on your notation. The one I have seen the most is $\displaystyle \mathbb{S} ^n = \{ x\in \mathbb{R} ^{n+1} : \Vert x \Vert =1 \}$ in which case it is indeed true that $\displaystyle \mathbb{S} ^n \setminus {e_n} \cong \mathbb{R} ^n$
To prove it, try to generalize the proof of the stereographic projection used to build the Riemann sphere which can be found in any (good) book on basic complex analysis