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)