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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)

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    $\begingroup$ Aren't there $2^{{2^\kappa}}$ many non-homeomorphic spaces of size $\kappa$ and only $2^\kappa$ many equivalence relations? Then a counting argument would work. $\endgroup$
    – Henno Brandsma
    Commented Oct 23, 2014 at 20:53

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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.

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