The number of non-solutions to an equation in a group and   non-topologizable torsion-free groups
  It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of non-solutions to this equation. A countable torsion-free non-topologizable group is constructed. 
Anton A. Klyachko, Anton V. Trofimov