topological structure on smooth manifolds

Question

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?

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

Answer 1

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).

Answer 2

In differential geometry you have maps $\phi: R^N \to M^N$ i.e parametrizations and in differential topology you have maps $\psi: M^N \to \mathbb{R}^N$ i.e coordinate charts. All of these maps are diffeomorphisms (or homeomorphisms). Therefore, you can choose in the differential topology setting to look at $\psi^{-1}: R^N \to M^N$. I think the big thing is that differential topology adheres to the fact that parametrizations though they may exist, are hard to explicitly write down.

Just to make sure what I'm saying is not just being lost in conversation: consider the definition for a set $A$ in $M$ to be open (differential topology sense). Let $\{(U, \phi)\}$ be an atlas for $M$. The sets for which we know are open are those which are the domain of some map $\phi$. Notice that we can't say that there is a map $\phi: A \to \mathbb{R}^N$ since it is not the case that $A$ is contained in the domain of some chart. Therefore, we will have to settle with, $A$ being open if there exists a $U$ s.t $\phi: A \cap U \to \mathbb{R}^N$ is a homeomorphism (or diffeomeorphism). However, this adheres to the definition of open in the usual sense. We've called $A \cap U$ to be open, then for $a \in A$ we take $\mathcal{O} = U \cap A \subset A$ i.e for any $x \in A$ there exists an open set about $x$ which is contained in $A$.

The above handles the same problem in which Do'Carmo addresses. He first takes a parametrization $x_a: U_a \to M$. Then he considers $x_a(U_a) \cap A$ and says if $x^{-1}_a(x_a(U_a) \cap A)$ is open, then $A$ is open in $M$. If you treat $x_a^{-1} = \phi$, you have your connection.

In Lee's definition you take a manifold $M$. To talk about a topological structure on $M$, you begin by using the fact that it is locally Euclidean i.e $U$ open in $M$ then there exists a homeomorphism $f_a: U_a \to \mathbb{R}^N$ for some $N$ and every $a \in M$. Therefore, if you like, you can take open sets of $M$ to be of the form $f_a^{-1}(W_a)$ where $W_a \subset U_a$ open. Now take $T = \{f_a^{-1}(W_a): a\in M\}$ then $(M,T)$ is a topological space i.e $M$ is a topological manifold.

To put more structure on $(M,T)$ we require more on the transition maps i.e of $f_i,f_j$ overlap in domain, if we demand that $f_i \circ f_j^{-1}$ and $f_j \circ f_i^{-1}$ are smooth then we've required that the images of the overlap $U_{ij}$ be diffeomorphic i.e $f_i(U_{ij})$ is diffeomorphic to $f_j(U_{ij})$ which is highly desirable. Consider integrals in multivariable-calculus. With this requirement, we can always apply the change-of variables formula! i.e made integration over $f_i(U_{ij})$ is messy, then we can change over to $f_j(U_{ij})$ and apply the formula with the map $F = f_{j} \circ f_i^{-1}$.