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.