1. The number of points plus 1 which can be removed from a curve without separating the curve into more than one piece.

2. The number of closed cuts or cuts joining points of previous cuts (or joining points on the boundary) plus 1 which can be made on a surface without separating the surface. Also known as Betti number.

3. In general, the n-dimensional connectivity number of a topological space X is the number of infinite cyclic groups whose direct sum with the torsion group G_n(X ) forms the homology group H_n(X ).