TITLE: topological structure on smooth manifolds
QUESTION [5 upvotes]: In John Lee's Introduction to Smooth Manifolds, a smooth manifold is defined as a topological manifold with a smooth structure. In do Carmo's Riemannian Geometry, a differentiable (smooth) manifold is defined by giving the smooth structure on merely a set $M$ and the author makes a remark that such smooth structure induces a natural topology on $M$.
Here is my question:

In Lee's definition, what is the relation between the topological structure and the smooth structure? Must the topology of the manifold (in Lee's definition) induced by the smooth structure in the way that do Carmo mentions?


The following are the definition and the remark by do Carmo mentioned above.

REPLY [2 votes]: There are two aspects:
First, the two definitions are easily not the same, since in Lee's definition of a topological manifolds it is assumed that the topology is Hausdorff and second countable. Thus the two standard examples: The line with two origins (Non Hausdorff) and any uncountable set with the discrete topology (Non second countable) are smooth manifold in DoCarmo's definition, but not in Lee's definition. 
Second, the above is the only difference. If we assume further that in DoCarmo's definition the topology induced on the set $M$ is both Hausdorff and second countable, then $M$ with the topology given is a topological manifold, and the atlas $\{ (x_\alpha^{-1}, x_\alpha (U_\alpha)\}_{\alpha}$ is a smooth structure (as in Lee's definition) on $M$. 
Remark: Though Lee's definition is the "morally correct" one, in practical situation people almost always use DoCarmo's convention $x_\alpha : U_\alpha\to M$ to perform local calculations (one cannot do any meaningful local calculation on $x_\alpha(U_\alpha)$, which is just an open set of an abstract topological manifold).