Weak regularity and consecutive topologizations and regularizations of pretopologies

AbstractL. 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)kρ>(RT)nρ for each k<n and (RT)nρ 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 ⩾ω0. Moreover, all these pretopologies have the property that all the points except one are topological and regular

SEQUENTIAL ORDER OF PRODUCT-SPACES

AbstractWe study the sequential order of product spaces. In some classes of sequential spaces we show the product theorems for sequential order. We construct under the continuum hypothesis two Fréchet spaces whose product is sequential and its sequential order is ω1

Consonance and Cantor set-selectors

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.peer-reviewe