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Betti number
An important topological property of a surface, named after the Italian mathematician Enrico Betti (1823-1892). The Betti number is the maximum number of cuts that can be made without dividing the surface into two separate pieces. If the surface has edges, each cut must be a "crosscut," one that goes from a point on an edge to another point on an edge. If the surface is closed, like a sphere, so that it has no edges, each cut must be a "loop cut," a cut in the form a simple closed curve. The Betti number of a square is 0 because it is impossible to crosscut without leaving two pieces. However, if the square is folded into a tube, its topology changes – it now has two disconnected edges – and its Betti number changes to 1. A torus, or donut shape, has a Betti number of 2.
Related entry
• chromatic number
Related categories
• TYPES OF NUMBERS
• MATHEMATICS
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