Charts on Topological Manifolds - Simple Notational Issue

[Math Amateur]

I am reading "An Introduction to Differential Topology" by Dennis Barden and Charles Thomas ...

I am focussed on Chapter 1: Differential Manifolds and Differentiable Maps ...

I need some help and clarification on an apparently simple notational issue regarding the definition of a chart (Definition 1.1.3) ...

Definition 1.1.3 reads as follows:

My question regarding this definition is as follows:

What is the meaning of M and how does it differ from M^m?

Surely the relationship between M and M^m is not the same as the relationship between R and R^m ... ?

I am not even sure what M is ... ?

Can someone clarify the above issue for me ...?

Hope someone can help ...

Peter===========================================================

So that readers can understand the context and notation of Barden and Thomas, I am providing the pages of the text leading up to and including the definition referred to above ... ... as follows ... ...

 

[fresh_42]

$M^m$ is just a reminder that the manifold $M$ is of $m$ dimensions. So $M^m = M$.
I have never seen such a notation but it's clear from definition 1.1.1. It only symbolizes that the (Euclidean) coordinates are in $ℝ^m$, i.e. there are $m$ coordinates. I guess the author drops the $m$ in $M^m$ and sticks with $M$ when he doesn't need to emphasize the dimension.

 

[Math Amateur]

Thanks fresh_42 ... appreciate the help ...

Peter