Heredity and cartesian closed coreflective subcategories of the category of topological spaces

The category of finitely-generated spaces is shown to be the largest hereditary cartesian closed coreflective subcategory of the category Top of topological spaces. Consequently, any coreflective subcategory of Top which is contained in a cartesian closed coreflective subcategory and contains a space which is not finitely-generated fails to be hereditary.