On Topological Congruences of a Topological Semigroup

We continue the study of the topological Rees congruences on a topological semigroup. First, we give some necessary and sufficient conditions for the question “when the quotient space of S over one of its ideal I is a kw-space?”, then we use it to generalize the Rees version of Lawson and Madison’s well-known theorem about topological congruences on S.