Description of the topological universe hull

Abstract

The topological universe hull of a concrete c-category over Set is compared with the cartesian closed topological and extensionable topological hulls. It is shown that the topological universe hull can be obtained from the other two hulls by a two-step process, in which the order of formation of hulls cannot be reversed. This provides also a correction of an existing description of the topological universe hull.

The quasitopos hull of the construct of closure spaces

“…A if B is a finally dense extension of A with the property that any finally dense embedding of A into a quasitopos can be uniquely extended to B. F. Schwarz [21] proved that the quasitopos hull of a construct (if it exists) can be described as the cartesian closed topological hull of the extensional topological hull. Using this we have the following proposition.…”

“…It is well known that the construct Top of topological spaces and continuous maps is not a quasitopos: it is neither extensional nor cartesian closed. Schwarz [21] showed that the topological quasitopos hull of a construct -if it exists-can be described as the cartesian closed topological hull of the extensional topological hull. Therefore we first construct the extensional topological hull of Cls and then the cartesian closed topological hull of this extensional hull.…”