On categories of ordered sets with a closure operator
We de ne and study two categories of partially ordered sets endowed with a closure ::: operator. The rst category has order-preserving and continuous maps as morphisms ::: and it is shown to be concretely isomorphic to a category of ordered sets endowed with ::: a compatible preorder. The second category has closed maps as morphisms and it is ::: proved to be cartesian closed. Consequences of these results for categories of closure ::: spaces and, in particular, of topological spaces are discussed.
Abstract We characterize those ordinals ξ for which Cp(ξ) is a weakly Whyburn space. As a byproduct we obtain the coincidence of the Frechet property and the Whyburn one on Cp(ξ).