Topology is the study of the properties of spaces that are insensitive to continuous deformations. Physicists recognize several different subdivisions of topology. The simplest is called point set topology, which is needed to define the notion of a topological space itself. A topological space is a set of points, possibly continuously infinite in number, which is endowed with a topology, or a collection of open sets that in essence specifies which points in the space are close to each other. This is the minimum mathematical structure needed to define continuous functions from one topological space to another. Continuous functions are simply those functions that take points that are near each other in one space to points that are near each other on the target space. An example of a family of continuous functions is polynomials in one variable which map the real line back to itself.

Two topological...