(Exercise 2.27 Lee’s Topological Manifolds) Let $C$ be the set $$C =\{(x,y,z)\mid \max\{|x|,|y|,|z|\}=1\}.$$ Show that the continuous map $\varphi :C\to\mathbb{S}^2$ defined by $$\varphi(x,y,z) = \frac{(x,y,z)} {\sqrt{ x^2+y^2+z^2}},$$ is a homeomorphism by showing that its inverse is given by $$\varphi^{-1}(x,y,z) = \frac{(x,y,z)}{\max\{|x|,|y|,|z|\}} \\ = (x,y,z),$$ a continuous map.
My confusion is this: obviously, we need to show that $\varphi\circ\varphi^{-1} = \text{id}:C\to\mathbb{S}^2$, although I’m not sure how this is defined since, in each argument, there would be three spacial variables making this a huge headache (as long as this definition is synonymous with the definition of composition for funtions $\Bbb R\to\Bbb R$). What does he mean?
He means the usual composition. As for the inverse: Let $(x,y,z) \in S^2$, then: $$ \varphi^{-1} (x,y,z) = \frac{(x,y,z)}{\max{|x|,|y|,|z|}} $$ Let $m$ be the max in the denominator: Applying $\varphi$ to this, we have: $$ \varphi \circ \varphi^{-1} (x,y,z) = \varphi \left(\frac{(x,y,z)}{m} \right) = \frac{(x,y,z)}{m} \left( \frac{x^2}{m^2}+ \frac{y^2}{m^2}+ \frac{z^2}{m^2}\right)^{-1/2} = \frac{(x,y,z)}{\sqrt{x^2 + y^2 +z^2}}$$ But since $x,y,z$ are on the sphere, $x^2 + y^2 +z^2=1$, so: $$\varphi \circ \varphi^{-1} (x,y,z) = \frac{(x,y,z)}{1} = (x,y,z) $$ So this is the identity. As for dealing with a mess of variables, such is life I suppose.