Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces.
In the present interesting paper the authors show that a space is Hausdorff-reducible if and only if it is hereditarily irresolvable, and offer several equivalent characterizations of hereditarily irresolvable and scattered spaces. They also show that for filtral topologies each of the three notions of submaximal, hereditarily irresolvable and Hausdorff-reducible coincide with the fact that the nonempty open sets form an ultrafilter. The authors construct by expanding some examples by El'kin, Padmavally and Anderson a compact irreducible \(T_1\)-space and a connected Hausdorff space, each of which is strongly irresolvable. Finally, the authors prove that the three notions of scattered, Hausdorff-reducible and hereditarily irresolvable coincide for a large class of spaces, including metric, Alexandroff, first countable, locally compact Hausdorff and spectral spaces.
scientific article; zbMATH DE number 1974780