Coreflectively modified duality in convergences and convergence appproach spaces
by
Frédéric Mynard
Burgundy University, France

The equivalence between the exponentiality of an object X in a bireflective subcategory R (with reflector R) of a cartesian-closed topological category C and the commutation problem (1) is known from F. Schwarz [6].

(1)     for allY in Ob(C),    X×R Y >= R(X×Y).

(\theequation)

Moreover, the link between exponentiality of X and quotientness (in R) of Id_X ×f for every quotient map f (in R) is well known (see for example [5]), so that (1) applies to problems of preservation of quotientness under product.

On the other hand, the coarsest C-structure \theta on a C-object X for which

(2)     for allY in Ob(C),    \theta×Y >= R(X×Y),

(\theequation)

is the reflection of X in the cartesian-closed hull of R. A common generalization of (1) and (2) is

(3)     for allY in Ob(C),    \theta×L Y >= R(X×Y),

(\theequation)

where L is another bireflector in C.

Applied in the category Conv of convergences, this general scheme (3) allows a unified treatement of many problems concerning product of most of the classically used types of quotient maps in general topology (quotient, hereditarily quotient, countably biquotient, biquotient, almost open...). Relativizing (3) to Y in a bicoreflective subcategory of C, several problems of preservation under product of many topological properties such as sequentiality, Fréchetness, strong Fréchetness, k-ness, quasi-k-ness and countably bi-k-ness (among others) can be handled simultaneously. In this way, classical results are unified and refined, new ones are obtained and open problems are solved. See [1], [2], [3]. On the other hand, extending some of the techniques developed in Conv to CAP, the theory applies in convergence-approach spaces. In particular, it gives a new point of view on exponential objects in the category PRAP of pre-approach spaces. See [4].

[]

S.  Dolecki and F.  Mynard. Convergence-theoretic mechanism behind product theorems. to appear in Top. Appl., 2000.

[]

F.  Mynard. Strongly sequential spaces. Comment. Math. Univ. Carolinae, 41:143-153 , 2000.

[]

F.  Mynard. Coreflectively modified continuous duality applied to classical product theorems. to appear, 2000.

[]

F.  Mynard. Abstract coreflectively modified duality and applications to convergence-approach spaces. to appear, 2000.

[]

F. Schwarz. Powers and exponential objects in initially structured categories and application to categories of limits spaces. Quaest. Math., 6:227-254, 1983.

[]

F. Schwarz. Product compatible reflectors and exponentiality. In Proceedings of the international conference held at the university of Toledo, Bentley & Al eds., Heldermann, Berlin, 1983.

Date received: June 5, 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-35.