The Rees-Suschkewitsch Theorem for simple topological semigroups

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup.