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I'd like to show that the sphere

[tex]\mathbb{S}^n := \{ x \in \mathbb{R}^{n+1} : |x|=1 \}[/tex] is the one-point-compactification of [tex]\mathbb{R}^n[/tex] (*)

After a lot of trying I got this function:

[tex]f: \mathbb{S}^n \setminus \{(0,...,0,1)\} \rightarrow \mathbb{R}^n [/tex]

[tex](x_1,...,x_{n+1}) \mapsto (\frac{x_1}{1-x_{n+1}},...,\frac{x_n}{1-x_{n+1}}) [/tex]

This is a continuous function, its image is the whole [itex]\mathbb{R}^n[/itex]. If I find its inverse [itex]f^{-1} [/itex] now and show that this one is continuous as well with [itex]image(f^{-1})=\mathbb{S}^n \setminus \{(0,...,0,1)\}[/itex] I have shown (*).

But I don't find the inverse. [itex] y_i=\frac{x_i}{1-x_{n+1}} [/itex] so [itex] x_i=(1-x_{n+1})*y_1 [/itex] but there is no [itex]x_{n+1}[/itex] here y is a n-dimensional vector...?

How can I find the inverse of f?

Regards

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# Finding inverse for a homeomorphism on the sphere (compactification)

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