Metric Space - Generalizations of Metric Spaces

Generalizations of Metric Spaces

• Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.

• If we consider the first definition of a metric space given above and relax the second requirement, we arrive at the concepts of a pseudometric space or a dislocated metric space. If we remove the third or forth, we arrive at a quasimetric space, or a semimetric space.

• If the distance function takes values in the extended real number line R∪{+∞}, but otherwise satisfies all four conditions, then it is called an extended metric and the corresponding space is called an -metric space. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of generalized ultrametric.

• Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.

• A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.