I recently learned about this in a topology class but wanted to know how to apply the Euler Characteristic for surfaces such as projective planes and Klein bottles (nonorientable surfaces).
I have come across a few proofs with different approaches for the Euler Characteristic for orientable surfaces where you start with $χ(Σ_g): = faces − edges + vertices = 2 − 2g. $
Following similar logic from those proofs, I tried to see if I could come up with a proof for nonorientable surfaces (mostly by using induction) and by starting with $χ(Σ_k): = 2 − k $
(from Wiki, don't quite understand why this is the way it is).
So, what would the proof for Euler’s Characteristic for nonorientable surfaces look like? Thanks.
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