Some cartesian closed topological hulls in approach theory.
by
Mark Nauwelaerts
University of Antwerp

Although being topological is a nice property for a construct, it may be desirable to have more convenient properties, such as being cartesian closed topological (CCT), which means that it is possible to equip the set of functions between two objects with some appropriate structure in such a way that one obtains canonical function spaces having some nice properties. Considering a category A that lacks this property, one can look for its CCT hull B, that is, the smallest possible extension of A that has this required property.

For example, in [2], G. Bourdaud indicated the existence of a "family" of cartesian closed topological constructs in CONV, the category of convergence spaces and continuous maps, such that the CCT hull of TOP and the CCT hull of CREG are specific instances of this family.

Inspired by a CCT hull in [1], it is shown in [5] that such a family of CCT constructs also arises in a (quasi-)distance-like setting, i.e. in pqsMET^\infty, the category of extended pseudo-quasi-semi-metric spaces and nonexpansive maps (also known as qDist), and is such that some CCT hulls arise as special instances of this family.

The previous convergence-like and (quasi-)distance-like settings were unified by E. Lowen and R. Lowen in CAP, the category of convergence-approach spaces and contractions, which was introduced in [3] as a topological universe extending AP, the category of approach spaces and contractions introduced earlier as a unification of topological and (extended pseudo-)quasi-metric concepts, and containing the subcategory UAP of AP consisting of uniform approach spaces, which is a unification of completely regular topological spaces and (extended pseudo-)metric spaces.

In [7], a family of CCT subconstructs in CAP is introduced such that restricting it down (in some sense) to the previously mentioned settings yields the aforementioned families, and such that some CCT hulls ([4], [6]) again arise as particular instances of this family.

References.

[1] J. Adámek and J. Reiterman, Cartesian closed hull for metric spaces,
Comment. Math. Univ. Carolinae 31, 1 (1990), 1-6.

[2] G. Bourdaud, Some cartesian closed topological categories of convergence spaces,
in: E. Binz and H. Herrlich (eds.), Categorical Topology (Proc. Mannheim 1975), Lecture Notes Math. 540, Springer, Berlin et al. (1976), 93-108.

[3] E. Lowen and R. Lowen, A quasitopos containing CONV and MET as full subcategories,
Intern. J. Math. & Math. Sci. 11 (1988), 417-438.

[4] E. Lowen, R. Lowen and M. Nauwelaerts, The cartesian closed hull of the category of approach spaces,
to appear in: Cahiers Topol. Géom. Diff. Cat.

[5] M. Nauwelaerts, Cartesian closed hull for (quasi-)metric spaces (revisited),
to appear in: Comment. Math. Univ. Carolinae.

[6] M. Nauwelaerts, The hulls of the category of uniform approach spaces,
submitted for publication.

[7] M. Nauwelaerts, Some cartesian closed topological constructs of convergence-approach spaces,
submitted for publication.

Date received: May 24, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-12.