Axioms of separation in semitopological groups and related functors

Abstract

We prove that for every semitopological group G and every i∈{0,1,2,3,3.5}, there exists a continuous homomorphism φG,i:G→H onto a Ti (resp., Ti&T1 for i⩾3) semitopological group H such that for every continuous mapping f:G→X to a Ti- (resp., Ti&T1- for i⩾3) space X, one can find a continuous mapping h:H→X satisfying f=h∘φG,i. In other words, the semitopological group H=Ti(G) is a Ti-reflection of G. It turns out that all Ti-reflections of G are topologically isomorphic. These facts establish the existence of the covariant functors Ti for i=0,1,2,3,3.5, as well as the functors Reg and Tych in the category of semitopological groups and their continuous homomorphisms.We also show that the canonical homomorphisms φG,i of G onto Ti(G) are open for i=0,1,2 and provide an internal description of the groups T0(G) and T1(G) by finding the exact form of the kernels of φG,0 and φG,1. It is also established that the functors Reg and Ti∘T3, for i=0,1,2 are naturally equivalent.