Abstract In this paper we continue our considerations of algebraic categories of spaces [8,9]. Especially, various separation and compactness properties are investigated. We disprove the wide-spread feeling, that there are only trivial examples besides categories of compact Hausdorff spaces. In fact, every functor F : Set→ Top, which is left adjoint to the underlying set functor F ( Set)→ Set, serves as left adjoint for the underlying set functor of some algebraic subcategory of Top. Hence free spaces are free algebras! However, the Sierpinski space as well as the reals are never contained in any algebraic category of spaces. But there is an algebraic category of Hausdorff spaces cogenerated by the first uncountable ordinal space. This one admits no varietal extension in Top. The trivial and the compact cases are characterized by mild additional assumptions, generalizing results in [5,8,9]. For instance, the only closed hereditary, productive, and varietal subcategories of Top are the categories Comp 2 of compact Hausdorff, Ind of indiscrete, and Triv of at most one point spaces.