Topology

Abstract

In several theorems of analysis and geometry the connectivity of a figure may play an essential part; for instance, the simple theorem that a function is one-to-one if its derivative is everywhere non-zero is only valid if the domain of definition of the function is connected. It is easy to find a function that is defined on the open intervals (0,1) and (2, 3), is not one-to-one, but has the derivative 1 at every point of the intervals (Fig.). The situation is more complicated when connectivity in the plane is discussed. The theorem that a vector field with zero curl has a potential holds, in general, only if the domain where the field is defined contains no holes; such a domain is called simply-connected. For example, one can choose the lengths of the vectors of a vector field in such a way that the curl becomes zero (Fig.), although it has no potential; this can be seen by integrating along the curve marked in the illustration. Investigations of the connectivity properties of figures suggested by these and similar examples form a small but characteristic part of topology.