How rare are egalitarian binary relations?

Knoblauch (2014) and Knoblauch (2015) investigate the relative size of the collection of binary relations with desirable features as compared to the set of all binary relations using symmetric difference metric (Cantor) topology and Hausdorff metric topology. We consider Ellentuck and doughnut topologies to further this line of investigation. We report the differences among the size of the useful binary relations in Cantor, Ellentuck and doughnut topologies.