Topological difference of the iterated remainders

Repeat taking remainders of Stone-Čech compactifications of the rationals Q, the irrationals P, and the Sorgenfrey line S,Q⁎=βQ\Q,Q⁎⁎=βQ⁎\Q⁎,Q⁎⁎⁎,⋯;P⁎,P⁎⁎,P⁎⁎⁎,⋯;S⁎,S⁎⁎,S⁎⁎⁎,⋯. We show that all of these spaces are topologically different; the case of Q answers van Douwen's question [2].