A surface is a two-dimensional topological manifold. A closed surface is a surface without boundary.

A result called the “classification theorem” gives us a symbolic semantics, matching the geometrical view point, in terms of genera, orientability and number of boundary components. Together with the connected sum operation, they make available a powerful language to be explored and exploited.

As an example of a surface take $T=S^{1}\times S^{1}$ the two torus, the boundary of a solid sugar donut shaped cake $D^{2}\times S^{1}$, where $S^{1}$ is the familiar modulus one complex numbers.