If (X,Ƭ) is a topological space, then a quasi-uniformity U on X is compatible with Ƭ if the quasi-uniform topology, Ƭ_u = Ƭ. This paper is concerned with local properties of quasi-uniformities on a set X that are compatible with a given topology on X.

Chapter II is devoted to the construction of Hausdorff completions of transitive quasi-uniform spaces that are members of the Pervin quasi-proximity class.

Chapter III discusses locally complete, locally precompact, locally symmetric and locally transitive quasi-uniform spaces.

Chapter IV is devoted to function spaces of quasi-uniform spaces.

Chapter V and the Appendix are concerned with the topological homeomorphism groups of quasi-uniform spaces.