Cartesian closedness, extensionality and bornological universes (Q2014747)

A topological space endowed with a bornology is called a \textit{bornological universe}. A closed bornological universe is a bornological universe where each bounded set is contained in a closed, bounded set. The category of all closed bornological universes and bounded continuous maps is a topological construct. Similarly, a bornological universe in which each bounded set is contained in an open, bounded set is called an open bornological universe but now the category of all open bornological universes and bounded continuous maps is not a topological construct. This paper provides a description of the cartesian closed topological hull, extensional topological hull and topological universe hull of both the category of closed and of open bornological universes. In the former case, the topological universe where the hulls are constructed is the supercategory \textbf{PsBorUniv} of all pseudobornological universes and bounded continuous maps, in which the category of closed bornological universes is fully embedded as a finally dense subcategory. In the latter case, the open bornological universes are not finally dense in the category of pseudobornological universes. Now the final hull of the category of open bornological universes is the full subcategory of \textbf{PsBorUniv} that contains all objects in which each convergent ultrafilter contains a bounded set. This is the topological universe where the hulls are constructed in the open case.