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.



Is there an uncountable, non-discrete, Hausdorff Toronto space?
You can read *"The Toronto problem"* by William Rea Brian (February 2014) to learn pretty much everything that is known about this problem. The article includes proofs of the folklore facts mentioned by Apollo and some other very interesting facts (e.g. Kunen´s result: An uncountable Hausdorff Toronto space contains no non-trivial convergent sequences).


But to sum up, the problem is now as open as it was 24 years ago: Under $GCH$ the only Hausdorff Toronto spaces (of any cardinality) are the discrete spaces. Also under $PFA$ there are no $T\_3$ non-discrete Toronto spaces of size $\aleph\_1$. We don´t know if the existence of a Hausdorff non-discrete Toronto space is consistent.


