Consider a topological space $X$. Lee in Introduction to Smooth Manifolds wrote that it is impossible to introduce a smooth structure on the topological manifold based only on topology (i.e. preservable my homeomorphisms) since the square and the circle are homeomorphic, but the square we don't want to be smooth.
Anyway, when introducing topological/smooth manifolds one makes a strict connection with $\mathbb R^n$. It seems that we extremely need this space to introduce the notion of smoothness on other spaces, am I right? Namely, can we formulate the smoothness without talking about $\mathbb R^n$ at all? If yes, what is necessary to have: finite dimension? metrics? topology?
Best Answer
Instead of introducing a smooth structure on the manifold $X$ by specifying a smooth structure using a "smooth" atlas, you could try instead to specify the set of "smooth" functions $X\to\mathbb{R}$. For example, the set of "smooth" functions for a compact submanifold is the restriction of the smooth functions of the embedding manifold to the submanifold. The problem is that this also works for non-smooth manifolds like the square, so you now need criteria how to distinguish non-smooth manifolds like the square from smooth manifolds like the circle, based on this set of "smooth" functions $X\to\mathbb{R}$. And even before that, you need criteria that such an "arbitrary" set of smooth functions must satisfy.
It might be fun trying to get such an approach to work. But at least for smooth manifolds, analytic manifolds and complex manifolds, the approach with the atlas is preferable. For differentiable manifolds, there might be questions where approaches like the one sketched above add value.