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Tu Manifolds Section 5.1

Definition of locally Euclidean of dimension n.

Example

To show $varphi_R: U setminus p to B setminus 0$ is also a homeomorphism, I think:

1. 
   $varphi_R = r circ varphi circ j$ where $j: U setminus p to U$ is inclusion and $r$ is some function $r: B(0,varepsilon) to B(0,varepsilon) setminus 0$, $varphi_R$ is bijective because $varphi(p)=0$ and $varphi$ is bijective and $varphi_R$ is continuous if we can find a continuous $r$ because $j$ and $varphi$ are continuous. Then $varphi_R$ is a homeomorphism if we can find an $r$ and show either that $varphi_R$ is an open or closed map or that $varphi_R^{-1}$ is continuous.

Is that correct?

2. 
   

   What $r$ can we select?

   
   
   
   
   

   • Must $r$ be a retraction? That is, is it insufficient to find a continuous $r$ that isn't a retraction, if it's possible?

   
   

   • Is it possible? (to find a continuous $r$ that isn't a retraction)

   
   
   
   


3. 
   

   Why is $varphi_R^{-1}$ continuous?

   
   
   
   
   
   • What is $varphi_R^{-1}$ anyway? I don't think $j^{-1}$ or $r^{-1}$ exist.
   
   
   
   



4. What properties of open or closed maps are there to say $varphi_R$ is an open or closed map because $varphi$ is an open or closed map and some other facts?

We have $varphi(x)=varphi_R(x)$ for all $x in U setminus p$, so I think $varphi_R(x)$ is an open map too. This might be relevant:

Theorem 22.1 in Munkres: