Gary Gruenhage - More on a-Toronto spaces

A Toronto space is an uncountable non-discrete Hausdorff space which is homeomorphic to every one of it's uncountable subspaces. It is known that a Toronto space, if one exixts, is scattered of Cantor-Bendixon rankw₁ with each level countable. J. Steprans defines an a-Toronto space, where a is an ordinal, to be a scattered space of rank a which is homeomorphic to each subspace of the same rank. We will discuss results related to our recent proof that, consistently, there are countable a-Toronto spaces for any a < w₁; for example, we show that the proof can be modified to obtain, for any cardinal k, a k-Toronto space in which each level has cardinality k.