Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of $ P_R^2 $ but I do not feel comfortable working with the projective plane, still not, and we've never seen anything more about this. But I can use the theorems of chapter 1 of massey on triangulations, about the Euler characteristic is independent of the triangulation, and so on. The problem is this :
First let's define a line as a set of the form $ L_{a,b,c} = \left\{ {\left[ {x,y,z} \right] \in P_R^2 :ax + by + cz = 0} \right\} $ Now let's define an array of lines, as a finite collection of lines $ \left\{ {L_i } \right\}_{i = 1}^n $ such that $ \bigcap\limits_{i = 1}^n {L_i } $ it´s empty . A paving of polygons of $ {P_R^2 } $ R is the same definition of "on triangles", but now you can consider any polygons. Prove that the Euler characteristic of $ {P_R^2 } $ R can be calculated with any paving of polygons $ \begin{align*} & v - \ell + p \\ v &= \text{number of vertices}\\ \ell &= \text{number of sides}\\ p &= \text{number of polygons} \end{align*} $ How can i prove this? I can use the theorem that the number is independent of the triangulation. It should not be at all difficult, but I'm a little unsure )= in the formalism. I think that I must triangulate each polygon and, and in someway count have v-l + p using the triangles sorry for the stupid question