Metrically generated theories

Authors:
E. Colebunders and R. Lowen

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1547-1556

MSC (2000):
Primary 54B30, 18B99, 18E20

DOI:
https://doi.org/10.1090/S0002-9939-04-07633-6

Published electronically:
November 19, 2004

MathSciNet review:
2111956

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Many examples are known of natural functors describing the transition from categories of generalized metric spaces to the ``metrizable" objects in some given topological construct . If preserves initial morphisms and if is initially dense in , then we say that is -metrically generated. Our main theorem proves that is -metrically generated if and only if can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

**1.**Jiří Adámek, Horst Herrlich, and George E. Strecker,*Abstract and concrete categories*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. The joy of cats; A Wiley-Interscience Publication. MR**1051419****2.**G. Choquet,*Convergences*, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.)**23**(1948), 57–112. MR**0025716****3.**E. Lowen and R. Lowen,*Topological quasitopos hulls of categories containing topological and metric objects*, Cahiers Topologie Géom. Différentielle Catég.**30**(1989), no. 3, 213–228 (English, with French summary). MR**1029625****4.**E. Colebunders, R. Lowen and M. Sioen. Saturated collections of metrics, preprint.**5.**D. Dikranjan and W. Tholen,*Categorical structure of closure operators*, Mathematics and its Applications, vol. 346, Kluwer Academic Publishers Group, Dordrecht, 1995. With applications to topology, algebra and discrete mathematics. MR**1368854****6.**A. Frölicher and A. Kriegl.*Linear Spaces and Differentiation Theory*, Pure and Applied Mathematics, John Wiley & Sons, 1988. MR**0961256 (90h:46076)****7.**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition; Graduate Texts in Mathematics, No. 43. MR**0407579****8.**Horst Herrlich,*A concept of nearness*, General Topology and Appl.**4**(1974), 191–212. MR**0350701****9.**Hans-Peter A. Künzi,*Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology*, Handbook of the history of general topology, Vol. 3, Hist. Topol., vol. 3, Kluwer Acad. Publ., Dordrecht, 2001, pp. 853–968. MR**1900267****10.**R. Lowen,*Approach spaces*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1997. The missing link in the topology-uniformity-metric triad; Oxford Science Publications. MR**1472024****11.**S. A. Naimpally and B. D. Warrack,*Proximity spaces*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. MR**0278261****12.**G. Preuss.*Theory of Topological Structures*, Mathematics and its Applications, Kluwer Academic Publishers, 1988. MR**0937052 (89m:54014)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54B30,
18B99,
18E20

Retrieve articles in all journals with MSC (2000): 54B30, 18B99, 18E20

Additional Information

**E. Colebunders**

Affiliation:
Vrije Universiteit Brussel, Vakgroep Wiskunde, Pleinlaan 2, 1050 Brussel, Belgium

Email:
evacoleb@vub.ac.be

**R. Lowen**

Affiliation:
Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium

Email:
bob.lowen@ua.ac.be

DOI:
https://doi.org/10.1090/S0002-9939-04-07633-6

Keywords:
Topological construct,
topological space,
metric space,
uniform space,
approach space,
bornological space,
measurable space

Received by editor(s):
September 22, 2003

Received by editor(s) in revised form:
January 5, 2004

Published electronically:
November 19, 2004

Communicated by:
Alan Dow

Article copyright:
© Copyright 2004
American Mathematical Society