Distinguish the homeomorphism from the imbedding in the definition of locally Euclidean in Tu Manifolds

Question

Tu Manifolds Section 5.1

Definition of locally Euclidean of dimension n.

A proposition in Section 5.2

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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.

1. Am I correct?

2. Is $j \circ \varphi$ equal or somehow equivalent to $\varphi$ for the same reason we have these equivalent definitions of local Euclidean?

3. 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 $\mathbb R^n$?

Answer 1

Regarding your questions:

1) Yes, this is the definition of being a homeomorphism onto an open subset ($W \subset \mathbb{R}^n$), but your notation is off, what you need to write down is that $\varphi: U \to \mathbb{R}^n$ factors as $\iota \circ \varphi'$, where $\iota$ is the inclusion, i.e. $\iota \circ \varphi'=\varphi$ with $\varphi'$ an homeomorphism. Actually, modulo some technicallies, "every subset is actually an injective map, and every injective map corresponds to an inclusion". Hence the notion of subset is often replaced by an inclusion.

2)well, the thing you mention here uses that every Ball in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$ (nocanonical), and that you can restrict your charts to only hit balls (since the charts are homeomorphisms). So no, the non canonicality of this crashes an equivalnece between $\varphi' and \varphi$.

3)This boils down again to the technicality that $j \circ \varphi = \varphi$ does not make sense (even for domain reasons). I think all of these question should be kind of clear if you wrap your hand around this technicallity of distinguishing between factoring and being equal!