Quasipseudometric Spaces and Topological Properties

In this paper, we introduce a new concept of metric: quasipseudometric, by relaxing the usual axiomatic of metrics. We study the consequences from a theoretical point of view. In particular, we focus ourselves on the ways to associate general topological structures to such a quasipseudometric. We show we get a weakened version of a topology: a pretopology, which can however be endowed with interesting properties.