We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a “limit function” λ which with each filter ℱ associates a map λ ℱ from the underlying set to the extended positive real line. Continuous maps and contractions can both be characterized as limit function preserving maps.

The properties common to both the convergence and metric case serve as a basis for the definition of the category, CAP. We show that CAP is a quasitopos and that, apart from the categories CONV, of convergence spaces, and MET, of metric spaces, it also contains the category AP of approach spaces as nicely embedded subcategories.