We prove that for many well known spaces A (such as n-spheres (n not 1, 3 or 7), or a one-point join of two spheres, or a Klein bottle, etc.), and for all but the most trivial equational theories T, the theory T cannot be modeled by continuous operations on T. This is, algebraically, the best possible generalization of known facts - e.g. that the aforementioned spheres cannot be H-spaces.
Date received: January 11, 1999.