Hausdorff properties of topological algebras

Abstract

Let P be a property of topological spaces. Let [P] be the class of all varieties \( \scr V \) having the property that any topological algebra in \( \scr V \) has underlying space satisfying property P. We show that if P is preserved by finite products, and if \( \neg P \) is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition.¶The property that all T0 topological algebras in \( \scr V \) are j-step Hausdor. (Hj) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to \( T_0 \Rightarrow H_j \) by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.¶Finally, we show that the topological implication \( T_0 \Rightarrow H_j \) holds in every k-permutable, congruence modular variety.