Orderconvergence, a quasitopos for Bounded Topology (BT)

Orderconvergence provides a unifying framework for the study of point convergence, boundedness, and filter merotopy, and plays a central role in both Convenient Topology and Bounded Topology. It offers a flexible approach to convergence phenomena that extends beyond the limitations of purely topological methods. In this paper, we investigate spaces equipped with order convergence from a categorical point of view and analyze the structural properties of the categories naturally associated with them.

It is well known that several classical categories arising in topology and convergence theory fail to satisfy desirable convenience properties, such as being Cartesian closed, extensional, or stable under the formation of quotients and products. In particular, the category TOP lacks these structural features, while the category CHY of Cauchy spaces, although Cartesian closed, does not form a quasitopos. Moreover, the full subcategory EF-PROX of CHY, consisting of Efremovič proximity spaces, also fails to possess the expected convenient properties.

To address these shortcomings, we introduce and study an appropriate supercategory of order convergence spaces together with suitable morphisms. We show that this category forms a strong topological universe in the sense of Convenient Topology, and in particular a quasitopos. As a consequence, quotients in this setting are stable under arbitrary products. This categorical framework enables a uniform treatment of bornological and Cauchy structures and provides a clearer understanding of their mutual relationships and categorical behavior. To exclude pathological cases, all underlying sets are assumed to be non-empty.