The Origin of Generalized Metric Spaces

In this chapter, we derive most of the generalized metric spaces that will be discussed in this book from three perspectives: distance functions, bases or their extensions and generalized countable compactness. At the same time, we give some simple characterizations of these spaces, describe their basic operation properties, such as hereditary properties, productive properties and properties preserved under topological sums, and discuss their preliminary relationships with some other classes of spaces.