I've been working through an exercise in Lee's Smooth Manifold. The author ask us to show that the space $\bar{\mathbb{R}}_+^n := [0,\infty)^n$ is homeomorphic to upper half-space $\mathbb{H}^n := \mathbb{R}^{n-1}\times[0,\infty)$, both equiped with the usual topology.
I managed to construct a homeomorphism $f : \bar{\mathbb{R}}_+^2 \to \mathbb{H}^2$ defined by restriction of the map $f(z) = z \, e^{i\theta} = r \, e^{i2\theta}$ to $\bar{\mathbb{R}}_+^2=[0,\infty) \times [0,\infty) \subset \mathbb{R}^2 \approx \mathbb{C}$. Alternatively, a more pleasing the map like $f(z) = z^2$. So we have $$ \mathbb{H}^2\approx \bar{\mathbb{R}}^2_+. $$ To proceed i tried for $n=3$ case but it's messy. After a while i came up with this induction: Suppose that $\mathbb{H}^n\approx \bar{\mathbb{R}}^n_+$ for all dimension less than or equal to $n$. Clearly this hold for $n=1$. By definition of upper half-space, \begin{align} \bar{\mathbb{R}}^{n+1}_+ = \bar{\mathbb{R}}^{n}_+ \times \bar{\mathbb{R}}_+ &\approx \mathbb{H}^n \times \bar{\mathbb{R}}_+ = \mathbb{R}^{n-1} \times \bar{\mathbb{R}}_+ \times \bar{\mathbb{R}}_+ \\ &\approx \mathbb{R}^{n-1} \times \mathbb{H}^2 = \mathbb{R}^{n-1} \times \mathbb{R} \times \bar{\mathbb{R}}_+ = \mathbb{R}^n \times \bar{\mathbb{R}}_+ = \mathbb{H}^{n+1}. \end{align} In the argument above i use the following result : if $A \approx B$ then $A \times C \approx B \times C$, which i think can be done by set $f : A \times C \to B \times C$ to be $f(a,c) = (\varphi(a),c)$, where $\varphi$ is a homeomorphism between $A$ and $B$.
Is this proof correct ? Is there any other (correct) proof for this ? I tried to googling it but i found nothing. If possible, can anyone give me references for manifold with corner ? Thank you.