I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows: (these are not Milnor's words so please let me know if something is unclear)

He first defines a smooth manifold as a subset of $R^A$ where $A$ can be any index set by means of charts: ($R^A$ is topologized as a Cartesian product)

A subset $N \subset R^A$ is a smooth manifold of dimension $n$ if each point in $M$ has a neighborhood diffeomorphic to an open set in $R^n$ such that in each point the tangent space be $n$-dimensional (equivalently the transition functions be smooth).

Also a smooth function on $N$ can be defined using charts.

Then he defines a smoothness structure on a set:

Let $M$ be a set and let $F$ be a collection of real valued functions on $M$ which separates points (that is, given $x \neq y $ in $M$ there exists $f\in F$ with $f(x) \neq f(y)$ ). Then $M$ can be identified with its image under the canonical imbedding $i :M \rightarrow R^F $. The collection $F$ is a smoothness structure on $M$ if the subset $i(M) \subset R^F$ is a smooth manifold, and if $F$ is precisely the set of all smooth real valued functions on this smooth manifold.

This definition is confusing for me and I don't know how to work with it, can anyone give me some intuition on this and clarify me or give me a reference? Maybe it's strange for me because he defines smoothness structure on a set not on a topological space.

This is one of the problems in this chapter about smoothness structure:

Let $P^n$ be the set of all lines through the origin in $R^{n+1}$ and let $q:R^{n+1}-{0} \rightarrow P^n$ be the quotient map. Let $F$ denote the set of all functions $f:P^n \rightarrow R$ such that $f \circ q$ is smooth. Show that F is a smoothness structure on $P^n$.

Also it's not clear why this is equivalent to usual smooth structure on real projective space which we talk about it.

sorry for long question.


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