Algebra Universalis, 47 (2002), 343-366

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 $\neg P$ 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 ($\uH_j$) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to $T_0\Rightarrow \uH_j$ 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
$T_0\Rightarrow T_2$ holds in every $k$-permutable, congruence
modular variety.