Can the set of all sigma-algebras over an arbitrary set be uncountable?

Solution 1:

Based on a comment "If it is so that this property can be extended to uncountable intersections..." I finally see the problem.

Fact 1. If $\mathcal F$ is a sigma algebra, $E_1,E_2,\dots\in\mathcal F$, and $E=\bigcap_{j=1}^\infty E_j$ then $E\in\mathcal F$.

Fact 2. If $\Omega$ is a set, $S$ is any nonempty collection of sigma algebras on $\Omega$ and $\mathcal F=\bigcap_{A\in S}A$ then $\mathcal F$ is a sigma-algebra.

No, Fact 1 cannot be extended to uncountable intersections. This is not contradicted by Fact 2, because Fact 2 is not an extension of Fact 1! The two facts are talking about different things.