I study simplicial homology topic in algebraic topology, where I read about the triangulation of compact matritizable spaces and I am going to compute the triangulation of real projective plane RP².
My question is, can't we take the following triangulation as a triangulation of RP² plane? https://i.stack.imgur.com/L1n1S.jpg
I think, for the two 2-simplexes <A,E,B> and <B,E,G> of the following triangulation, $$<A,E,B> \cap <B,E,G> = <B> \cup <E> $$ which is not a simplex at all, and so it violates the conditions for a set of simplexes to become siplicial complex.Please, clear my doubt.
No, you cannot make such a triangulation, because there are 2 triangles with the same vertices: $A, B, E$.
I am not a specialist either, I am currently studying algebraic topology, too; but it is very clear that this is not allowed. At least in the usual way of defining triangulations: my book says there are other ways with relaxed rules that allow this (but probably have some drawbacks, such as making some proofs more difficult).