Editing Euler characteristic (section)

===Betti number alternative===
More generally still, for any [[topological space]], we can define the ''n''th [[Betti number]] ''b''<sub>''n''</sub> as the [[rank of an abelian group|rank]] of the ''n''-th [[singular homology]] group. The '''Euler characteristic''' can then be defined as the alternating sum

:<math>\chi = b_0 - b_1 + b_2 - b_3 + \cdots.</math>

This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index&nbsp;''n''<sub>0</sub>. For simplicial complexes, this is not the same definition as in the previous paragraph but a homology computation shows that the two definitions will give the same value for <math>\chi</math>.