The answer is no, by the argument given by Henno in his comment.
It true that there are exactly $2^{2^\kappa}$-many non-homeomorphic topologies on a set of size $\kappa \geq \aleph_0$. See for example [this][1] nice argument by Stefan Geschke. On the other hand it is obvious that there are only $2^\kappa$-many binary relations on a set of size $\kappa\geq \aleph_0$, so it follows that there are no infinite cardinals for which (S) is true.  


  [1]: https://math.stackexchange.com/a/65742