Betti number

Etymology
A calque of, coined in 1892 by ; named after Italian mathematician in recognition of an 1871 paper.

Noun

 * 1)  Any of a sequence of numbers, denoted bn, which characterise a given topological space K by giving, for each dimension, the number of holes in K of said dimension;  the rank of the nth homology group, Hn, of K.
 * 2) * 1979 [W. H. Freeman & Company], Michael Henle, A Combinatorial Introduction to Topology, 1994, Dover, page 163,
 * Prove that, for compact surfaces, the zeroth Betti number is the number of components of the surface, where a component is a connected subset of the surface, such that any larger containing subset is not connected.
 * 1) * 2007, Oscar García-Prada, Peter Beier Gothen, Vicente Muñoz, Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles,, page 7,
 * P ROPOSITION  2.1. Fix the rank $$r$$. For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers.
 * P ROPOSITION  2.1. Fix the rank $$r$$. For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers.

Usage notes

 * The dimensionality of a hole (as used in the definition) is that of its enclosing boundary: a torus, for example, has a central 1-dimensional hole and a 2-dimensional hole (a "void" or "cavity") enclosed by its ring.
 * Informally, the Betti number $$b_n$$ represents the maximum number of cuts needed to separate K into two pieces ($$n$$-cycles).
 * $$b_0$$ can be interpreted as the number of components in $$K$$.

Translations

 * Danish: Bettital
 * French: nombre de Betti
 * German: Bettizahl
 * Italian: numero di Betti