Weak regularity and consecutive topologizations and regularizations of pretopologies

Abstract

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n < ω there is a (regular) pretopology ρ (on a set of cardinality c) such that ( RT ) k ρ > ( RT ) n ρ for each k < n and ( RT ) n ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ⩾ ω 0 . Moreover, all these pretopologies have the property that all the points except one are topological and regular.