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Knoblauch (2014) and Knoblauch (2015) investigate the relative size of thecollection of binary relations with desirable features as compared to the setof all binary relations using symmetric difference metric (Cantor) topology andHausdorff metric topology. We consider Ellentuck and doughnut topologies tofurther this line of investigation. We report the differences among the size ofthe useful binary relations in Cantor, Ellentuck and doughnut topologies. Itturns out that the doughnut topology admits binary relations with more generalproperties in contrast to the other two. We further prove that among theinduced Cantor and Ellentuck topologies, the latter captures the relative sizeof partial orders among the collection of all quasi-orders. Finally we showthat the class of ethical binary relations is small in Ellentuck (and thereforein Cantor) topology but is not small in doughnut topology. In essence, theEllentuck topology fares better compared to Cantor topology in capturing therelative size of collections of binary relations.

How rare are the properties of binary relations?