Quasi-metric and Quasi-uniform Spaces

Abstract

The first chapter of the book is concerned with the topological properties of quasimetric, quasi-uniform, asymmetric normed and asymmetric locally convex spaces. A quasi-metric on a set X is a positive function ρ on X × X satisfying all the axioms of a metric excepting symmetry: it is possible that ρ(y, x) ≠ ρ(x, y) for some x, y ∈ X. Similarly, a quasi-uniformity is a family ${\mathcal U}$ of subsets of X × X satisfying all the requirements of a uniformity excepting symmetry: U ∈ ${\mathcal U}$ does not imply that U ⁻¹ = {(y, x) : (x, y) ∈ U} belongs to ${\mathcal U}$. The lack of symmetry in the definition of quasi-metric spaces and of quasi-uniform spaces causes a lot of troubles, mainly concerning completeness, compactness and total boundedness in such spaces. There are several notions of completeness in quasi-metric and quasiuniform spaces, all agreeing with the usual notion of completeness in the case of metric or uniform spaces, each of them having its advantages and weaknesses. Also, countable compactness, sequential compactness and compactness do not agree in quasi-metric spaces, in contrast to the metric case. In spite of these differences, there are a lot of positive results relating compactness, completeness and total boundedness in quasi-metric and in quasi-uniform spaces which are presented in this chapter. A quasi-metric ρ and its conjugate ρ(x, y) = ρ(y, x), x,y ∈ X, generate in the usual way two topologies τ_ρ and τρ, that makes X a bitopological space, with a similar situation for quasi-uniform spaces. For this reason we have included in this chapter a quite detailed study of bitopological spaces including pairwise separation properties, Urysohn and Tietze type theorems for semi-continuous functions and Baire category.