It is a well-known result in the study of topological groups that any T0-topological group is also regular and satisfies the stronger separation axiom T3. The same holds for topological quasi-groups. In the area of universal algebra, the only result on condition T3 is a negative one due to Coleman, showing that congruence permutability is not strong enough to force the implication T0=>T3 to be valid for the topological algebras of a variety.
In this paper, we will provide similar negative results for a large class of varieties. As a consequence, we can conclude that if the implication T0=>T3 holds in a non-trivial variety V , each defining set of equations for V must contain at least one equation with an instance of a funtion symbol in the position of an argument to another such instance.