International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 3, Pages 417-438
doi:10.1155/S016117128800050X
A quasitopos containing CONV and MET as full subcategories
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.
Abstract
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 λℱ 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.