Sigma-algebra for countable sample space Let $E=\{\{\omega \} : \omega \in \Omega \}$. 
$\sigma ( E) = \{ A \subseteq \Omega : A \ is \ countable \  or \  A^c \ is \  countable \} $ is a $\sigma$-algebra generated by the set $E$ i.e. the smallest $\sigma$-algebra containing $E$ (already proved).
Prove:
$\sigma (E)$ is equal to partitive set of $\Omega$ if and only if $\Omega$ is a countable set.
One side: Suppose $\Omega$ is countable. $\sigma (E)$ contains all the sets that are countable or their complement is countable. That is true for every subset of a countable set, so it leads $\sigma (E)$ is a partitive set of $\Omega$.
Other side: please help

If $\Omega$ is not countable then it can be written as disjoint union of two uncountable subsets $\Omega_1,\Omega_2$.
See here for a proof of that.
So then $\Omega_1,\Omega_2\notin \sigma(E)$.
