Partial Metric Spaces

As with many mathematical concepts these axioms are chosen to ensure that two things are equal if and only if some property expressible in terms of the concept holds. For a metric space, x = y if and only if d(x, y) = 0. Thus as there is the equality relation x = y in a metric space, so there is what we call an indistancy relation d(x, y) = 0. Axioms M1 and M2 work together to identify equality with indistancy. That is x and y are equal if and only if x and y have no distance between them. Such identification may seem to be so fundamental that to suggest otherwise would serve no purpose. However, there is a longstanding precedent for relaxing the axioms which ensure this identification. The relation defined by x ≡ y if and only if d(x, y) = 0 is an equivalence, which can be useful, as in the construction of the classical l p-spaces. In this construction, spaces are considered in which axiom M2 is dropped while the others hold, giving a pseudometric space. This article retains M2 but drops M1, introducing the possibility of equality without indistancy, and leading to the study of self-distances d(x, x) which may not be zero. Originally motivated by the experience of computer science, as discussed below, we show how a mathematics of nonzero self-distance for metric spaces has been established, and is now leading to interesting research into the foundations of topology. The approach of this article is to retrace the steps of a standard introduction to metric and topological spaces [10], seeing why and how it can be generalized to accommodate nonzero self-distance. The article then concludes with a discussion of research directions. Proofs of results presented here consist of straightforward reasoning about distances or topology, and as such are left as informative exercises for the reader. For more material and publications please visit [7], the authors’ web site partialmetric.org