International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 3, Pages 417-438

A quasitopos containing CONV and MET as full subcategories

E. Lowen1 and R. Lowen2

1University of Brussels, V.U.B., Departement Wiskunde, Pleinlaan 2, Brussels 1050, Belgium
2University of Antwerp, R.U.C.A., Wiskundige Analyse, Groenenborgerlaan 171, Antwerp 2020, Belgium

Received 12 May 1987; Revised 12 November 1987

Copyright © 1988 E. Lowen and R. Lowen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 associates a map λ 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.