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.