Books where the concept of smooth manifold is defined for arbitrary sets (not topological spaces)

Question

I am looking for books where the concept of smooth manifold is defined for arbitrary sets; i. e., books beginning with the definition of chart as a one-to-one map with open image and moving on to the definition of manifold as set endowed with a maximal atlas. This one would be perfect if didn't have so many missing pages.

Answer 1

I don't see how one can define a smooth manifold for an arbitrary set.

When you say

"beginning with the definition of chart as a one-to-one map with open image"

to speak about subsets of a set being open, we need to have a topology on that set, and then that set with a topology is exactly a topological space.

A smooth manifold $M$ is basically a topological manifold equipped with a smooth structure. For $M$ to be a smooth manifold it first needs to be a topological manifold, that is for each $x \in M$ there must exist an open set $U$ containing $x$ such that $U$ is homeomorphic to an open subset of $\mathbb{R}^n$.

So if we started of with some arbitrary set $A$, we need to endow it with a topology, then show that $A$ possesses the property that for each $a \in A$ there exists a open set $V$ of $A$ containing $a$ that is homeomorphic to an open subset of $\mathbb{R}^n$. Then only once this is done can we talk about it being a smooth manifold or not.