Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi\_1,\phi\_2$ on $\mathbb{R}^n$ and on $\mathbb{R}^m$, respectively, such that
$$
\sup\_{x \in K}\left\|
\phi(x,y) -(\phi\_1(x),\phi\_2(y))
\right\|<\epsilon?
$$


In other words, is every homeomorphism on a product space approximately the product of homeomorphisms on its components?



Is every homeomorphism approximately a product of homeomorphisms?
Take $m=n=1$, $K=[0, 1]$ and $\phi$ to be rotation by 90 degrees.


Then the $x$-coordinate of $(\phi\_1(x), \phi\_2(y))$ is bounded for $x\in K$ and any $\phi\_1, \phi\_2$ while there are points with arbitrarily large $y$-coordinate (which becomes arbitrarily large $x$-coordinate upon applying $\phi$).


