The cluster set. Definition

The notion of a cluster set was first formulated explicitly by Painlevé ([1], p. 438) in his well-known Stockholm lectures of 1895 on differential equations. Painlevé introduced the cluster set, which he called domaine d'indétermination, as a descriptive notion to characterize in an intuitive way the behaviour of an analytic function in the neighbourhood of a singularity in terms of the properties of the set of all its limits at the singularity, and to classify the singularities of a function in terms of these cluster sets.

Although Painlevé was considering only analytic functions, the notion of a cluster set does not depend, any more than does the notion of a limit, upon the function being analytic or even continuous; nor need it be restricted to complex-valued functions of a complex variable. It is equally applicable to real functions of a real variable and to mappings in spaces of higher dimensions, as well as to general topological spaces, although not necessarily equally fruitfully in all cases. We shall therefore introduce the definitions in a sufficiently general form to include the special cases we shall require in later chapters.

Let the function w = f(z) be defined in a domain D which it maps into the w-sphere S. Unless otherwise stated, we shall take D to be in the z-plane, but if D is an open set on the real line, we shall assume the mapping to be into the real line or into a great circle in S.