Axioms of separation in paratopological groups and reflection functors (Q465863)

The author studies \(T_i\)-reflection functors for \(i=1,2,3\) and the reflection \(Reg(G)\) where \(G\) is a paratopological group. He first proves that for any paratopological group \(G\), the group \(T_3(G)\) coincides with the semiregularization of \(G\).NEWLINENEWLINEMoreover, the author proves that \(Reg(G)\) is the semiregularization of the group \(T_2(G)\), this result deduces a description of the paratopological group \(Reg(G)\). He observes that the functors \(Reg, T_3 \circ T_2, T_2 \circ T_3, T_1 \circ T_3\) and \(T_0\circ T_3\) are naturally equivalent in the category of paratopological groups.NEWLINENEWLINEFurthermore, he studies properties of the canonical homomorphism \({\varphi}_{G,r}: G \rightarrow Reg(G)\) and it turns out that \({\varphi}_{G,r}\) is a \(d\)-open mapping whenever \(G\) is a paratopological map.NEWLINENEWLINEFinally, the author proves that if \(H\) is dense subgroup of a paratopological group \(G\), then the image of \(H\) under \(T_3\) is topologically isomorphic to the subgroup \({\varphi}_{G,r}(H)\) to \(T_3(G)\) and, analogously, \(Reg(H)\) is topologically isomorphic to the subgroup \({\varphi}_{G,r}(H)\) of \(Reg(G)\). He also observes that these results cannot be extended for \(T_i\) where \(i=1,2\).