Is there an uncountable, non-discrete, Hausdorff Toronto space? 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.
 A: 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.
A: From "Open Problems in Topology" the following facts are known (described there as "folklore")


*

*any Hausdorff non-discrete Toronto space is scattered with countably many isolated points

*hence such a space must have derived length $\omega_1$ and be hereditarily separable, thus must be an $S$-space

*this gives a way to have a model where there are no non-discrete Hausdorff Toronto spaces of size $\aleph_1$: assume $2^{\aleph_0}\neq2^{\aleph_1}$ and note that hereditary separability implies that the space has only $2^{\aleph_0}$ autohomeomorphisms while any Toronto space of size $\lambda$ must have $2^\lambda$ autohomeomorphisms


I have no idea what has been proven since then.  It is also mentioned that questions concerning Toronto spaces with larger cardinalities and with stronger separation axioms is still open and gives for example the question "Are all regular (or normal) Toronto spaces of size $\aleph_1$ discrete?"
