Applied Categorical Structures 4: 69-79, 1996. 69 (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands. On the Largest Coreflective Cartesian Closed Subconstruct of Prtop E. LOWEN-COLEBUNDERS and G. SONCK* Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium *Aspirant NFWO (Received: 16 November 1994; accepted: 6 July 1995) Abstract. We show that the subconstruct Fing of Prtop, consisting of all finitely generated pre- topological spaces, is the largest Cartesian closed coreflective subeonstmct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated. We give a counterexample showing that without finite productivity the previous result does not hold. Mathematics Subject Classifications (1991). 54B30, 18D15, 54A05. Key words: pretopological space, finitely generated space, exponential object, Cartesian closed- hess. 1. Introduction It is well-known that the topological constructs Top, of topological spaces and continuous maps, and Prtop, of pretopological spaces and continuous maps, are not Cartesian closed. For Top, this fact was already observed in 1946 by Arens [2] and for Prtop the conclusion follows from essentially the same argument. Both these negative results were generalized by Schwarz [18] who proved that if an epireflective subconstmct of Prtop contains a non-indiscrete space, then it fails to be Cartesian closed. In this paper we will study Cartesian closedness for coreflective subconstructs of Prtop (containing at least one nonempty space). Some coreflective subcon- stmcts arise very naturally, in Top as well as in Prtop. We mention a few exam- pies. The functors F: L* -+ Prtop, mapping a sequential convergence space to the pretopological space on the same underlying set whose closure is derived from the convergent sequences, and G: L* --+ Top mapping a sequential convergence space to the topological space on the same underlying set whose closed sets are derived from the convergent sequences, both functors F and G leaving mor- phisms unchanged as set functions, have as image the coreflective subconstruct FrPrtop of Prtop of all Fr6chet pretopologies and the coreflective subconstruct SeqTop of Top of all sequential topologies, respectively ([7, 12]).