A Neometric Survey

Nonstandard analysis is often used to prove that certain objects exist, i.e., that certain sets are not empty. In the literature one can find many existence theorems whose only known proofs use nonstandard analysis; see, for example, [AFHL]. This article will survey a new method for existence proofs, based on the concept of a neometric space. We shall state definitions and results (usually without proofs) from several other papers, and try to explain how the ideas from these papers fit together as a whole. The purpose of the neometric method is twofold: first, to make the use of nonstandard analysis more accessible to mathematicians, and second, to gain a deeper understanding of why nonstandard analysis leads to new existence theorems. The neometric method is intended to be more than a proof technique— it has the potential to suggest new conjectures and new proofs in a wide variety of settings. However, it bypasses the notion of an internal set and the lifting and pushing down arguments which are the main feature of many nonstandard existence proofs. The central notion is that of a neocompact family, which is a generalization of the classical family of compact sets. A neocompact family is a family of subsets of metric spaces with certain closure properties. In applications, nonstandard analysis is needed at only one point—to obtain neocompact families which are countably compact. From that point on, the method can be used without any knowledge of nonstandard analysis at all. This program grew out of earlier work on adapted probability distributions ([K2], [HK]) and a first approach to neocompactness in the paper [K3]. Various aspects of our program will appear in the papers [CK], [FK1], [FK2], [FK3], [FK4], [K4], and [K5]. In this article we shall give an overview of the entire program. We shall explain how the method can be painlessly applied, and discuss the relationship of the method to nonstandard practice and to adapted probability distributions. Let’s take an informal look at a common way of solving existence problems in analysis (or in a metric space): We want to show that within a set C there exists an