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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 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.