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
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)