Motivation for Lee's definition of smooth map

Question

I am trying to understand why Lee makes the following definition for real valued smooth maps on manifolds.

Let M be a topological manifold. We call $f:M\to \mathbb{R}$ smooth if chart for $p \in M$, $(U, \varphi)$, satisfies the condition $f \circ \varphi^{-1} : \varphi(U) \to \mathbb{R}$ is smooth in the usual way.

What bugs me is that, definition says that function is smooth if behavior of it on a homeomorphic image of the local domain is smooth, but somehow it feels like instead of homeomorphic, the word should be diffeomorphic even though it is yet to be defined.

How does homeomorphism (the local coordinate map) not make 'smoothness' be lost a bit? I mean, how do we guarantee it? I suppose transition map being smooth was the way to guarantee it, but maybe I can get some more intuition into it.

PS: I guess one possible answer is that this is just a definition, but then how this was natural definition is not clear to me, as I would be puzzled seeing just availability of continuous map.

To add an example, I would like to think of graph of $\vert x\vert$ around $0$ as my manifold $M$, open space homeomorphic to $M$ as $(-1,1)$ and the map whose smoothness is to be determined to be $id$ and the local coordinate map to be projection to x coordinate $\pi$. I have seen some authors, e.g Kristopher Tapp, to define a map to be smooth on a closed set if it can be extended to a smooth map on a open set containing the initial domain. In this case $id$ is smooth on whole Euclidean space. But under this definiton $id$ is smooth iff $\pi^{-1}$ is smooth which is not.

Thus I am confused if there are many ways to capture 'smoothness' which are not necessarily same but depend on definition, which is a bit unsatisfactory since 'continuity' was captured by topology unambiguously.

Answer 1

The map $\varphi$ associated to a chart $(U,\varphi)$ is a diffeomorphism onto its image. However, one cannot say that before one has defined a smooth map (therefore before defining diffeomorphism)!

It seems to me that you are worried about the idea that the chart map need only be a homeomorphism, rather than a diffeomorphism. Let's take your example, the graph $\Gamma=\{(x,y)\in \mathbb{R}^2|y=|x|\}$. We can define a chart $(U,\varphi)$ with $U=\Gamma$ and $\varphi:(x,y)\mapsto x$. With this atlas, $\Gamma$ is diffeomorphic to $(-1,1)$.... But, $\Gamma$ is not diffeomorphic to $(-1,1)$, right? Well, here we need to be more careful. There are two different differential structures on $\Gamma$ in question:

1. there is the induced structure via the embedding of this graph into $\mathbb{R}^2$;
2. there is a structure induced by this chart.

These two are not the same differential structure, so they define different notions of smooth map on your manifold. In fact, according to the first, this is not a manifold!

Answer 2

Just to add to Joshua Tilley's explanation, the specific point of confusion for your question seems to lie in the statement:

"What bugs me is that, definition says that function is smooth if behavior of it on a homeomorphic image of the local domain is smooth"

NO, this definition is not saying that. It is saying that for an arbitrary chart $\phi$ for $p$, $f\circ\phi^{-1}$ will be smooth. The key distinction is that most homeomorphisms are not charts. Your smooth structure is defined by an arbitrarily chosen special class of mutually compatible homeomorphisms, wherein charts $\phi$ and $\psi$ are compatible with each other if $\phi\circ\psi^{-1}$ is smooth where it is defined.

Most local homeomorphisms will not be compatible with your chosen class of charts, and will therefore not be charts themselves. The definition is asking you only to ensure $f$ is smooth when composed with a chart.