Semicontinuous groups and separation properties.
The remarkable development of the theory of smooth quasigroups is surveyed.
It is proved, for various spaces A, such as a surface of genus 2, a figure-eight, or a sphere of dimension ≠ 1,3,7, and for any set Σ of equations, that Σ cannot be modeled by continuous operations on A unless Σ is undemanding (a form of triviality that is defined in the paper).