# Proving criterion establishing subset of euclidean space is smooth manifold

Our professor told us the following criterion:

Let $M\subseteq\mathbb{R}^k$ be any subset, and suppose that for all $m\in M$ we have a triplet $(U,\psi,A)$ with $m\in U\subseteq M$ an open subset, $\psi:A\to\mathbb{R}^k$ a smooth immersive map that, when its codomain is restricted to $U$, becomes a homeomorphism, where $A\subseteq\mathbb{R}^d$ is an open subset, then this $M$ is a smooth manifold, and the triplets form a smooth atlas on $M$.

I see how this is a topological atlas, but how do I prove the transition functions are smooth? I cannot use differentials since no differential here is invertible, so how can I proceed?

So we have $M\subseteq\mathbb{R}^k$ and we know that for all $m\in M$ we have an open $U\subseteq M$ (open in the subspace topology), an $A\subseteq\mathbb{R}^d$ for a fixed $d$, and $\psi:A\to U$ which is immersive as $A\to\mathbb{R}^k$, and a homeomorphism as $A\to U$. What we want to show is that the $\psi^{-1}$s are a smooth atlas for $M$.
So assume $\phi:U\to A$ and $\phi':U'\to A'$ are two of those homeomorphisms. What about $\phi\circ\phi'^{-1}$? Is it smooth? It is defined on $\phi'(U\cap U')$, and maps it homeomorphically onto $\phi(U\cap U')$. So let $p$ be a point of its domain. By the rank theorem, $\phi^{-1}$ is locally (perhaps on a subset of $\phi(U\cap U')$) represented by the inclusion, i.e. there is, on this subset of $\phi(U\cap U')$ containing $\phi\circ\phi'^{-1}(m)$, a change of coordinates such that with respect to the new coordinates one has $\phi(x_1,\dotsc,x_d)=(x_1,\dotsc,x_d,0,\dotsc,0)$. So if we compose the change of coordinates with the projection onto the first $d$ coordinate, i.e. consider $\pi_d\circ c$ with $c$ the coordinate change, we have a local inverse of $\phi^{-1}$, in other words, a smooth extension $\tilde\phi$ of $\phi$ to an open set in $\mathbb{R}^k$. Naturally, $\phi\circ\phi'^{-1}$ is, where it is defined, equal to $\tilde\phi\circ\phi'^{-1}$, but that is a composition of smooth functions, hence in the preimage of that set where $\phi^{-1}$ is the inclusion (the one given by the rank theorem) under $\phi\circ\phi'^{-1}$, the transition function is smooth. Such a procedure can be done for any point in the domain of the transition function, which is therefore smooth, proving our claim.