Page 1 Topology of Surfaces Topology is the abstraction of certain geometrical ideas, such as continuity and closeness. Roughly speaking, topology is the exploration of manifolds, and of the properties that remain invariant under continuous, invertible transformations, known as homeomorphisms. The basic problem is to classify manifolds according to homeomorphism type. In higher dimensions, this is an impossible task, but, in low dimensions, it can be done.
Surfaces are some of the simplest, yet most interesting topological objects.
Points of the first type are called interior points, and those of the second type are called boundary points. The set of all boundary points constitutes the boundary of the surface. It consists of one or boundary components, each of which is homeomorphic to a circle.