# Do diffeomorphisms between arbitrary sets extend to ordinary diffeomorphisms between open sets?

Let $$E_i$$ be a $$\mathbb R$$-Banach space and $$\Omega_i\subseteq E_i$$. $$g:\Omega_1\to E_2$$ is called $$C^1$$-differentiable if $$g=\left.\tilde g\right|_{\Omega_1}\tag1$$ for some $$E_2$$-valued $$C^1$$-differentiable function $$\tilde g$$ on an open neighborhood of $$\Omega_1$$. Moreover, $$f:\Omega_1\to\Omega_2$$ is called $$C^1$$-diffeomorphism if

1. $$f$$ is a homeomorphism from $$\Omega_1$$ onto $$\Omega_2$$;
2. $$f$$ and $$f^{-1}$$ are $$C^1$$-differentiable.

So, if $$f:\Omega_1\to\Omega_2$$ is a $$C^1$$-diffeomorphism, then $$f=\left.\tilde f\right|_{\Omega_1}\tag2$$ for some $$E_2$$-valued $$C^1$$-differentiable function $$\tilde f$$ on an open neighborhood $$N_1$$ of $$\Omega_1$$ and $$f^{-1}=\left.\tilde g\right|_{\Omega_2}\tag3$$ for some $$E_2$$-valued $$C^1$$-differentiable function $$\tilde g$$ on an open neighborhood $$N_2$$ of $$\Omega_2$$.

Is it possible to choose $$N_1,\tilde f,N_2,\tilde g$$ so that $$\tilde f$$ is a $$C^1$$-diffeomorphism (in the ordinary sense) from $$N_1$$ onto $$N_2$$ and $$\tilde g=\tilde f^{-1}$$?

I think you need more assumptions. To be explicit, consider the $$x$$-axis in $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$. These are diffeomorphic, but clearly don’t have any diffeomorphic neighborhoods.