I want to know if there is a "natural" topology on $\mathbb{RP}^n$, if yes, how it is defined?
(a natural topology for me is a topology which, unless said differently, it is considered without said explicitly, for example, $\mathbb{R}^n$ and the euclidean topology, or the product topology on a product space)
To be more specific, suppose i have to calculate the homology groups (or fundamental group) for $\mathbb{RP}^n$. Clearly I have to choose two open sets with certain conditions.
My problem is I can't figure out what is open and what is not. (i'm interested in "viewing" the open sets on the quotient of $S^n$ and the other representations)
Another doubt is the construction of $\mathbb{RP}^n$. The quotient and CW-complex constructions are pretty clear, the one starting with an affine space not.
I tried to search around, and the wiki seems to have some answers, but i need some titles of books (undergraduate level) to understand better the topic.
Hope there is no similar question already asked.
Chapter 5 of Topology and Groupoids is on "Projective and other spaces" and deals simultaneously with the real, complex, and quaternionic projective spaces, and their cell structures. The details of the latter are on p. 150.
The calculation of fundamental groups is easy, since they are trivial except in the real case, when you get $\mathbb Z_2$ since there is a double covering map $S^n \to \mathbf{RP}^n, n >1$. You can also get this from the cell structure which for the real case starts off $e^0 \cup e^1 \cup e^2\cup \cdots $ where the attaching map of the $2$-cell is of degree $2$. For the homology of these spaces the results on cellular homology are needed, which I won't do here.