Spaces of Mappings and Spaces of Compact Subsets

Abstract

Usually in accounts of the theory of Ordinary Differential Equations in the description of topological properties of sets of solutions we use the notion of the uniform convergence of sequences of functions. We will discuss related topological structure in §§1 and 2. In fact its correspondence with this circle of questions is not very good because it deals with sets of functions with a fixed domain. In the definition of a solution of a differential equation we do not fix domains of definition. Functions with different domains are called solutions of the same equation. In order to describe topological properties of such sets of functions more completely, but rather briefly, we will introduce in §5 the space of partial mappings. In first few sections we consider some fundamental notions which are helpful in study of properties of the new space. See also [Ku1].