I am reading John M. Lee's book: Introduction to Smooth Manifolds ...

I am focused on Chapter 1: Smooth Manifolds ...

I need some help in fully understanding Example 1.3: Projective Spaces ... ...

Example 1.3 reads as follows:

My questions are as follows:

Question 1

In the above example, we read:

" ... ... define a map $ \displaystyle \phi_i \ : \ U_i \longrightarrow \mathbb{R}^n$ by

$ \displaystyle \phi_i [ x^1, \ ... \ ... \ , x^{n+1} ] = ( \frac{x^1}{x^i} , \ ... \ , \frac{x^{i-1}}{x^i} , \frac{x^{i+1}}{x^i}, \ ... \ , \frac{x^{n+1}}{x^i} )$

This map is well defined because its value is unchanged by multiplying x by a nonzero constant. ... ... "

Now, in the above, the domain of $ \displaystyle \phi_i$ is shown as an $ \displaystyle (n+1)$-dimensional point ... ... BUT ... ... $ \displaystyle \phi_i$ is a map with a domain consisting of lines in $ \displaystyle \mathbb{R}^{n + 1}$, so shouldn't the dimension of the domain be $ \displaystyle n$ ... ?

Maybe we have to regard the equivalence classes of the quotient topology involved as $ \displaystyle (n+1)$-dimensional points and recognise that points $ \displaystyle x = \lambda x$ where $ \displaystyle \lambda \in \mathbb{R}$ ... ... is that right?

(The statement about the map being well defined is presumably about recognising equivalence classes as on point in the projective space ... ... is that right?

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Question 2

In the above text from Lee's book we read:

"... ... Because $ \displaystyle \phi_i \circ \pi$ is continuous ... ... "

How do we know that $ \displaystyle \phi_i \circ \pi$ is continuous ... ?

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Question 3

In the above text from Lee's book we read:

"... ... In fact $ \displaystyle \phi_i$ is a homeomorphism, because its inverse is given by

$ \displaystyle {\phi_i}^{-1} [ u^1, \ ... \ ... \ , u^{n} ] = [ u^1, \ ... u^{i-1}, 1, u^i, \ ... \ , u^{n} ]$

as you can easily check ... ... "

I cannot see how Lee determined this expression to be the inverse ... why is the inverse of $ \displaystyle \phi_i$ of the form shown ... how do we get this expression ... and why is it continuous (as it must be since Lee declares $ \displaystyle \phi_i$ to be a homeomorphism ... ...

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Question 4

Just a general question ... in seeking a set of charts to cover $ \displaystyle \mathbb{RP}^n$, why does Lee bother with the $ \displaystyle \tilde{U_i}$ and $ \displaystyle \pi$ ... why not just define the $ \displaystyle U_i$ as an open set of $ \displaystyle \mathbb{RP}^n$ and define the $ \displaystyle \phi_i$ ... ... ?

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Hope someone can help with the above three questions ...

Help will be appreciated ... ...

Peter