A generalized distance based on a generalized triangle inequality

In this paper we present a new generalization of the mathematical notion of distance. It is based on the abstraction of the codomain of the distance function. The resulting functions must satisfy a generalized triangular inequality, which depends only on the order structure of the valuation space, i.e a monoid structure is not required. This type of functions will be called i-Distances (i-metrics, i-quasi-metric, etc.). We show that they generate a topology in a very natural way based on open balls. This paper generalizes (Santanaand Santiago, 2013) which has been successfully applied in the field of Clustering Algorithms (see Silva et al., 2014, 2015). An example in the field of Interval Mathematics is also investigated. The resulting topology is Hausdorf and regular but non-metrizable, what means that it cannot be generated by an usual metric.