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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

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I think your last comment is on the right track. Pick a polygon and triangulate it. How do $v$, $l$, and $p$ change? –  Adam Saltz Oct 4 '11 at 2:55
    
I did it! thanks for all –  August Oct 5 '11 at 1:51

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