Axioms of separation in paratopological groups and reflection functors

Abstract

We continue the study of Ti-reflections for i=0,1,2,3, as well as the regular reflection Reg(G) of a paratopological group G defined by the author earlier. We give ‘constructive’ descriptions of the paratopological groups Ti(G) for i=2,3 and Reg(G), for an arbitrary paratopological group G. It is known that the canonical homomorphism φG,i:G→Ti(G) is open for i=0,1,2 and φG,0 is perfect, for each semitopological group G. We show here that the homomorphisms φG,3:G→T3(G) and φG,r:G→Reg(G) are d-open, for every paratopological group G.It is also shown that if f:G→H is a closed (perfect) homomorphism of semitopological groups, then the homomorphisms Ti(f):Ti(G)→Ti(H) for i=0,1 are also closed (perfect). Similarly, if f:G→H is a perfect surjective homomorphism of paratopological groups, then the homomorphisms Ti(f) for i=2,3 and Reg(f) of Reg(G) to Reg(H) are also perfect. Finally, we prove that if H is a dense subgroup of a paratopological group G, then T3(H) is topologically isomorphic to the subgroup φG,3(H) of T3(G) and, similarly, Reg(H) is topologically isomorphic to the subgroup φG,r(H) of Reg(G). These results cannot be extended either to T1 or T2.