Applications of the reflection functors in paratopological groups

Abstract

We apply Ti-reflections for i=0,1,2,3, as well as the regular reflection defined by the author in [20] for the further study of paratopological and semitopological groups. We show that many topological properties are invariant and/or inverse invariant under taking Ti-reflections in paratopological groups. Using this technique, we prove that every σ-compact paratopological group has the Knaster property and, hence, is of countable cellularity.We also prove that an arbitrary product of locally feebly compact paratopological groups is a Moscow space, thus generalizing a similar fact established earlier for products of feebly compact topological groups. The proof of the latter result is based on the fact that the functor T2 of Hausdorff reflection ‘commutes’ with arbitrary products of semitopological groups. In fact, we show that the functors T0 and T1 also commute with products of semitopological groups, while the functors T3 and Reg commute with products of paratopological groups.