A note on the T2-reflection of semitopological groups

Given a semitopological group G, denote by N(e) the family of open neighborhoods of the identity e in G. We show that K = boolean AND{(U) over bar : U is an element of N(e)} is a closed invariant subgroup of G. Therefore, the T-2-reflection of G is topologically isomorphic to G/K. Also, we show that in every semitopological group G, the following conditions are equivalent: i) G is feebly compact; ii) T-0(G) is feebly compact; iii) T-1(G) is feebly compact; iv) T-2(G) is feebly compact. These results answer some questions posed by M. Tkachenko about the T-i-reflection of semitopological groups, for i = 1, 2. (C) 2014 Elsevier B.V. All rights reserved.