Class of all σ-fields containing a class

Question

given a set $ \Omega = \{0,1,2\}$ and a class $A = \{\{1\}\}$, what is the class of all σ-fields containing $A$

Methink it is $2^\Omega$ ?

Answer 1

No, $2^\Omega$ is the wrong kind of thing -- that's a set of sets of numbers, whereas a set of $\sigma$-algebras would be a set of sets of sets of numbers.

The correct solution cannot be written down quite as simply as your proposal, but on the other hand it's easy to work out explicitly by hand. One of the $\sigma$-algebras is (of course) $2^\Omega$ and there's only one other.