How to verify this given homeomorphism in Munkres on stereographic projection

Question

In munkres topology for theorem 59.3, he provided a homeomorphism between $S^n- p$ and $mathbb{R}^n$,
where
$$f(x) = frac{1}{1 - x_{n+1}} (x_1, dots , x_n)$$
and
$$f^{-1}(y) = (t(y)y_1, dots , t(y)y_n, 1- t(y))$$
given $t(y) = frac{2}{1+ ||y||^2}$.
how do I check that they are actually the inverse of each other?
(I tried to plug in one to the other, but he algebra is just horrible.)
(This is on page 369 in munkres)

Angina Seng · Accepted Answer

I dispute the "horror" of the algebra. For $f^{-1}(f(x))$, let's write $a$ for $1/(1-x_{n+1})$. Then $y_i=ax_i$ and
$$|y|^2=a^2|(x_1,ldots,x_n)|^2=a^2(1-x_{n+1}^2)=frac{1+x_{n+1}}{1-x_{n+1}}$$
and
$$1+|y|^2=frac2{1-x_{n+1}}$$
since $x$ is on the unit sphere. Then
$$t=t(y)=frac2{1+|y|^2}=1-x_{n+1}.$$
So
$$f^{-1}(y)=((1-x_{n+1})y_1,ldots,(1-x_{n+1})y_n,x_{n+1})=x$$
since $x_i=(1-x_{n+1})y_i$.
Doing $f(f^{-1}(y))$ is no more difficult.

How to verify this given homeomorphism in Munkres on stereographic projection

One Answer

Add your own answers!

Ask a Question