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 $(X,\tau)\cong (\kappa,\tau_\kappa)/\sim$.
Let $\kappa$ be an infinite cardinal. Let $X$ be a set of power $\kappa$, and let $\tau$ be a topology on $X$. There are $2^\kappa$ equivalence relations on $X$, so $\langle X,\tau\rangle$ has at most $2^\kappa$ distinct quotients.
For each $p\in\beta\kappa$ let $Y_p=\{p\}\cup\kappa$ with the topology that it inherits from $\beta\kappa$. For $p,q\in\beta\kappa$ the spaces $Y_p$ and $Y_q$ are homeomorphic iff the partial orders $\langle p,\subseteq\rangle$ and $\langle q,\subseteq\rangle$ are isomorphic, which is the case iff there is a bijection $\varphi:\kappa\to\kappa$ such that $q=\{\varphi[A]:A\in p\}$. There are $2^\kappa$ such bijections, but $|\beta\kappa|=2^{2^\kappa}$, so there are $2^{2^\kappa}$ pairwise non-homeomorphic spaces $Y_p$. Clearly almost all of them fail to be quotients of $\langle X,\tau\rangle$.