Hausdorff properties of topological algebras

Let P be a property of topological spaces. Let [P] be the class of all varieties V having the property that any topological algebra in V has underlying space satisfying property P. We show that if P is preserved by finite products, and if -P is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition. The property that all T-0 topological algebras in V are j-step Hausdorff (H-j) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to T-0 double right arrow H-j by showing that this topological implication holds in every (2(j) + 1)-permutable variety, but not in every (2j + 2)-permutable variety. Finally, we show that the topological implication T-0 double right arrow T-2 holds in every k-permutable, congruence modular variety.