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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 thisthis 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.

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.

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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Ramiro de la Vega
Ramiro de la Vega
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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.