The Toronto problem

A Toronto space is a topological space that is homeomorphic to every one of its full-cardinality subspaces, and the Toronto problem asks whether every Hausdorff Toronto space of size ℵ 1 is discrete. We examine compactness properties, convergence properties, and separation properties of non-discrete Hausdorff Toronto spaces of size ℵ 1 , and we classify the non- T 1 Toronto spaces of any infinite size.