Definition
For a non-negative integer k, the kth Betti number bk(X) of the space X is defined as the rank of the abelian group Hk(X), the kth homology group of X. Equivalently, one can define it as the vector space dimension of Hk(X; Q), since the homology group in this case is a vector space over Q. The universal coefficient theorem, in a very simple case, shows that these definitions are the same.
More generally, given a field F one can define bk(X, F), the kth Betti number with coefficients in F, as the vector space dimension of Hk(X, F).
Read more about this topic: Betti Number
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