Generated quasi-metric hyper and function spaces

In formal analogy to separable metric spaces we introduce the concept of a generated quasi-metric space. In a corresponding way as each point of a separable metric space can be represented as the limit of a sequence in some countable dense subset, each point of a generated quasi-metric space can be considered as the infimum of a sequence in the generating set (with respect to the partial order induced by the quasi-metric). Typically, the generating subset can be chosen such that it is itself a separable metric space (with respect to the metric induced by the quasi-metric). This concept enables a “countable access” to the points of the quasi-metric space and has some interesting applications in computer science. We prove that certain important hyper and function spaces can be naturally considered as generated quasi-metric spaces.