Editing Betti number (section)

== Formal definition ==
For a non-negative [[integer]]&nbsp;''k'', the ''k''th Betti number ''b''<sub>''k''</sub>(''X'') of the space ''X'' is defined as the [[rank of an abelian group|rank]] (number of linearly independent generators) of the [[abelian group]] ''H''<sub>''k''</sub>(''X''), the ''k''th [[homology group]] of&nbsp;''X''. The ''k''th homology group is <math> H_{k} = \ker \delta_{k} / \operatorname{Im} \delta_{k+1} </math>,  the <math> \delta_{k}</math>s are the boundary maps of the [[simplicial complex]] and the rank of H<sub>k</sub> is the ''k''th Betti number. Equivalently, one can define it as the [[vector space dimension]] of ''H''<sub>''k''</sub>(''X'';&nbsp;'''Q''') since the homology group in this case is a vector space over&nbsp;'''Q'''. The [[universal coefficient theorem]], in a very simple torsion-free case, shows that these definitions are the same.

More generally, given a [[Field (mathematics)|field]] ''F'' one can define ''b''<sub>''k''</sub>(''X'',&nbsp;''F''), the ''k''th Betti number with coefficients in ''F'', as the vector space dimension of ''H''<sub>''k''</sub>(''X'',&nbsp;''F'').