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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.

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 $mathbb R^n$?

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!