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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

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