On reflections of quasitopological groups and semitopological groups

Abstract

In this paper, we investigate the reflections of quasitopological groups and semitopological groups. We show that: (1) T3(G) is the semiregularization of G, i.e., T3(G)≅Gsr, for every quasitopological group G; (2) there is a quasitopological group G such that T1(T3(G))≇T3(T1(G)); (3) using the general stability theorems, many properties P are invariants and/or inverse invariants of φG,i for some i∈{0,1,2} in the category of semitopological groups; (4) by the construction of T3, we prove that many topological properties P are invariants and inverse invariants of φG,i for each i∈{3,r} in the category of semitopological groups, and give an affirmative answer to [53, Problem 4.2] for Reg(G); (5) if H is a dense subgroup of a semitopological group G, then the group T3(H) is topologically isomorphic to the subgroup φG,3(H) of T3(G) and Reg(H) is topologically isomorphic to the subgroup φG,r(H) of Reg(G), which gives a positive answer to [52, Problem 4.2].