On asymmetric distances (Q2851033)

The notion of asymmetric metric space as a set \(M\) equipped with a positive function \(b:M\times M\rightarrow\Re^{+}\) which satisfies the triangle inequality, but fails to be symmetric. The theory of length structure in the symmetric case is generalized. Also, the author generalizes the Busemann theory of general metric spaces, without using that the topology generated by the forwards balls and the backward balls coincide. This generalization has the following effect on the theory: There is no guarantee that an arc-parametrized rectificable path is continuous. In the second section, a review of the theory of length structures is given as well as some properties and examples that illustrate the theory. Furthermore, the author studies and compares three classes of paths. In the third section, the theory of asymmetric metric spaces is given and some general properties are studied too. Some classical examples are given. In Section 4 several examples are presented in order to illustrate the previous theory.