Cartesian closed coreflective subcategories of the category of topological spaces

AbstractAnswering the first part of Problem 7 in [10] we prove that there is no largest cartesian closed coreflective subcategory of the category Top of topological spaces and the same result for the category Haus of Hausdorff spaces. We also give a new simple proof of the known result (see [3]) that the category of all compactly generated spaces is not cartesian closed (here a compact space need not be Hausdorff). Moreover, new examples of cartesian closed coreflective subcategories of Top and Haus are presented