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=⋂{U¯:U∈N(e)} is a closed invariant subgroup of G. Therefore, the T2-reflection of G is topologically isomorphic to G/K. Also, we show that in every semitopological group G, the following conditions are equivalent: