From the textbook: Introduction to Differential Topology by TH. Brocker and K. Janich.
I'm trying to understand the identity atlas $\{ \textrm{Id}_U\}$.
Let's first look at how a manifold is defined:
Definition 1.1: An n-dimensional topological manifold $M^n$ is a Hausdorff topological space with a countable basis for the topology, which is locally homeomorphic to $\mathbb{R}^n$.
Now, looking at this Identity Atlas from example 1.5a:
1.5a: If $U \subset \mathbb{R}^n$ is an open subset, then the atlas ${\textrm{Id}_U}$, which only contains a single chart Id: $U \rightarrow U$ defines a differentiable atlas.
This confuses me, because I don't see how the dimensions can be correct. This could be key to my misconception, but I assume that a 2-manifold (which is a surface in $\mathbb{R}^3$) is locally homeomorphic to $\mathbb{R}^2$. Therefore $n=2$ here. However, I do not understand how this identity map is possible from $\mathbb{R}^3 \rightarrow \mathbb{R}^2$ as the authors say $U \rightarrow U$ meaning that both spaces are the same. I suspect that the fact that I assume we embed the manifold in $n+1$ is wrong as it is not specified in the book, I would appreciate clarification.