Keele Research Repository

The object of this thesis is to study topological properties of. correspondences, or set-valued mappings, and hyperspaces, i.e, spaces of subsets of given sets.
In chapter 1, we study correspondences under their purely set-theoretical aspect. We introduce canonical extensions and stress their usefulness in proving properties of correspondences.
Chapter 2 is devoted to topological structures on hyperspaces, including the finite topology and quasi-uniformities. Chapter 3 pursues that study. The unifying role of quasi-uniformities is discussed: not only do they often yield a common proof of different results, they also give way to new ones.
Finally, chapter 4 deals with so-called "covering topologies", such as the locally finite topology. We study their properties and.compare them with the uniform and the finite topologies.