Is there an infinite cardinal $\kappa$ for which the following statement (S) true?


(S) : There is a topology $\tau\_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq \kappa$ there is a binary relation $\sim$ on $\kappa$ such that $(X,\tau)\cong (\kappa,\tau\_\kappa)/\sim$.


(I'm transferring this question from <https://math.stackexchange.com/questions/987131/universality-with-respect-to-quotients>)



Universality with respect to quotients
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](https://math.stackexchange.com/a/65742) 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. 


