I am a little confused on what the Euler Characteristic of an n-sided spherical polygon would be. Because the polygon is 3 dimensional, would it be considered to have 2 faces or 1? Following that, I would think that the Euler Characteristic would simply be equal to the number of faces as the number of edges and vertices are the same.

  • $\begingroup$ The number of faces is indeed $2$, but not "because the polygon is 3 dimensional". It's because a polygon, as a simple curve, separates the sphere into two regions, each of which counts as a "face". So, you're correct, the Euler Characteristic reduces to the number of faces, confirming that the characteristic of the sphere is $2$. $\endgroup$
    – Blue
    Aug 21, 2018 at 7:31


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