# homeomorphism inbetween the $\mathbb{R}^n$ and an open unit cube

I want to find a homeomorphism that maps the open cube $W = (-1,1)^n\subseteq \mathbb{R}^n$ to the $\mathbb{R}^n$.

I know that these two are homeomorphic, but I don't know where to start when it comes to finding an actual function that satisfies the requirements for a homeomorphism inbetween these two.

• You should start with the case $n=1$. – Gregory Grant May 6 '15 at 15:16
• Well, in case $n = 1$, one could choose a function like $f(x) = tan\frac{π}{2}x$ (or a similar function that maps (-1,1) continuously onto $\mathbb{R}$. But how would this carry over to n > 1? – moran May 6 '15 at 15:19
• Right. Now just use the same function in each coordinate. – Gregory Grant May 6 '15 at 15:21
• If $f:(-1,1)\rightarrow\mathbb R$ is a homeomorphism, then $(x,y)\mapsto(f(x),f(y))$ is a homeomorphism of $(-1,1)^2$ to $\mathbb R^2$. – Gregory Grant May 6 '15 at 15:23
• Okay... well, is it that simple? So for $x = (x_1, ..., x_n)$, would $f(x) = (tan(\frac{π}{2}x_1), ..., tan(\frac{π}{2}x_n))$ already be a homeomorphism inbetween $(-1, 1)^n$ and $\mathbb{R}^n$? It's continuous and bijective as far as I can tell. And $f^{-1}(y) = (\frac{π}{2}artan(y_1), ..., \frac{π}{2}artan(y_n))$? – moran May 6 '15 at 15:28

Hint: Here is a start \begin{align}f_i: \mathbb R &\to (-1,1)\\x &\mapsto \frac{x}{1 + |x|}\end{align}
And use that each of the coordinate functions of $f = (f_1, f_2, \ldots, f_n)$ are continuous with continuous inverse.