We call a topological space $X$ a *Toronto space* if for any subspace $Y \subseteq X$ such that $Y$ and $X$ have the same cardinality it follows that $Y$ is homeomorphic to $X$.

Does anybody know what is known about the following question?:

Is there an uncountable, non-discrete, Hausdorff Toronto space?

It is not hard to show that if $X$ is countable, Hausdorff and Toronto then $X$ has the discrete topology. I have been thinking about the uncountable case for a while and it turns out it is a much harder question.

Partitioning topological spaces, by William Weiss, inMathematics of Ramsey theory: "[This] is of course related to the Toronto seminar problem of whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable subspaces. There are rules for working on this latter problem. The problem can be worked upon only in groups of three or more mathematicians, and it is required that alcohol, preferably beer, be present during this time. Contact anyone in the Toronto Set Theory Seminar for the current status of the problem. It may never be solved." $\endgroup$ – Andrés E. Caicedo Sep 24 '13 at 19:27