TOPOLOGIES ON SPACES OF SUBSETS BY ERNEST MICHAEL 1. Introduction. In the first part of this paper (§§2-5), we shall study various topologies on the collection of nonempty closed subsets of a topolog- ical spaced). In the second part (§§6-8), we shall apply these topologies to such topics as multi-valued functions and the linear ordering of a topological space. To facilitate our discussion, we begin by listing some of our principal con- ventions and notations: Convention 1.1. Let X be the "base" space. Then 1.1.1. An element of X will be denoted by lower case italic letters (for example, x). 1.1.2. A subclass of X will be called a set, and will be denoted by upper case italic letters (for example, E). 1.1.3. A class of sets will be called a collection, and will be denoted by light-face German letters (for example, 33). 1.1.4. A class of collections will be called a family, and will be denoted by bold face German letters (for example, 2t ). Convention 1.2. By a neighborhood of a class, we shall always mean a neighborhood of this class considered as an element of a topological space, not as a subclass of such a space. Notation 1.3. 1.3.1. A topological space X, with topology T, will be denoted by (X, T). 1.3.2. A uniform space X, with uniform structure U, will be denoted by [X, U]; the topology which U induces on X will be denoted by | U\. Notation 1.4. Let (X, T) be a topological space. Then tA(X) = IE CX | E is not empty}, 2X= {E(ZX\E is closed and not empty}, Jn(X) = {E£2-r|E has at most n elements}, J(X)={E^2x\E is finite}, Q(X) = {E G 2* | E is compact} (2). The following notation will be useful for defining and discussing our topologies : Notation 1.5. If { U, },ei is a collection of subsets of a topological space X, Presented to the Society, April 29, 1950; received by the editors April 26, 1950 and, in revised form, December 18, 1950. (') Such collections are often called hyperspaces. (2) Strictly speaking, we should write Q(X, T) and so forth, since Q(X) depends on T. Where no confusion can occur, we shall, however, adopt the simplified notation. 152 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use