Axioms of separation in semitopological groups and related functors

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