Tu Manifolds Section 5.1
Definition of locally Euclidean of dimension n.
A proposition in Section 5.2
For Definition 5.1, I think $j \circ \varphi: U \to \mathbb R^n$, where $j: W \to \mathbb R^n$ is inclusion and $W$ is an open subset of $\mathbb R^n$, is an imbedding while $\varphi: U \to W$ is the required homeomorphism.
Am I correct?
Is $j \circ \varphi$ equal or somehow equivalent to $\varphi$ for the same reason we have these equivalent definitions of local Euclidean?
For the proposition in Section 5.2, assuming I am correct that there is a difference between $j \circ \varphi$ and $\varphi$, is this difference irrelevant because $$(j \circ \varphi)(U \cap V) = \varphi(U \cap V)$$
where $j: W_U \to \mathbb R^n$ is inclusion and $\varphi: U \to W_U$ where $U$ is open in the topological manifold $M$, and $\varphi(U)=W_U$ is open in $\mathbb R^n$?