More on \alpha-Toronto spaces
by
Gary Gruenhage
Auburn University

An \alpha-Toronto space is a scattered space of Cantor-Bendixson rank \alpha which is homeomorphic to each of its subspaces of the same rank. For example, a convergent sequence is a 2-Toronto space. Last year we gave a consistent answer to a question of Steprans by constructing a model in which there exist \alpha-Toronto spaces for every \alpha < \omega₁. Here we review this construction, and extend it to show, e.g., that for \alpha < \omega₁ there can be \alpha-Toronto spaces of differing weights, and for each cardinal \kappa there can be a \kappa-Toronto space of cardinality \kappa. We also give a ZFC example of a 3-Toronto space. Finally, we mention several open problems.

Date received: June 21, 1998