Quantales and continuity spaces

The theory of metric spaces provides an elementary introduction to topology and unifies many branches of classical analysis. Using it, the intuitive notions of continuity, limit and Cauchy sequence can be developed in a very general setting. Moreover, since metric techniques are frequently much more powerful than topological ones, they make possible simpler and more elegant solutions to many problems. Unfortunately, not all topological spaces are metrizable, so these powerful techniques have limited applicability. This fact has led to a number of attempts to generalize the notion of metric space in order to extend the range of problems that can be treated with metric techniques. Some examples are: probabilistic metric spaces (Schweizer and Sklar [15]), Boolean metric spaces (Blumenthal [1], Chapter 15), Menger’s metrics for groups [10], and the structure spaces of Henriksen and Kopperman [4]. The many common properties shared by these examples and their similar underlying structures suggest the possibility of a general theory of metric spaces, which would include them, as well as ordinary metric spaces, as special cases. Kopperman’s theory of continuity spaces [7] and the work of Trillas and Alsina [16] are two notable efforts to provide such a theory. The natural starting point of any general theory of metric spaces is the choice of what are the essential properties of the structure ([0, ],5, +) from this point of view. These will certainly include the following: