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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
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—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)