Show that for any set Ω, the system
$$A=\{A ⊂ Ω: \textrm{A or } A^{c} \textrm{ is countable}\} $$ is a σ-algebra.
$$$$ So I couldn't show much here, except that I said that if $a∈A$ it implies that also $a^{c}∈A$. Moreover we know that if $A⊂ Ω$ but $A\neqΩ$ then, in order for the system to be a sigma algebra it needs to include the empty set and also the sample space. Also I said that: let $a_1,.....,a_i∈A$ this means that $\forall i∈\mathbb{N}: a_i∈A => \cup_{i∈\mathbb{N}}^{\infty}a_i∈A$.$$$$ Could somebody tell me if this is correct and if its not a way to prove it, could you tell me what can I do more?
What you're looking at is called the co-countable $\sigma$-algebra, say $\mathscr{A}$. Checking that $\emptyset, \Omega \in \mathscr{A}$ is simple enough, since $\emptyset$ is countable.
Given a countable collection of sets $A_n$ from $\mathscr{A}$, you have to show that their union $A = \bigcup_{n \in \mathbb{N}} A_n$ is in $\mathscr{A}$. This can be done by considering two cases: either all $A_n$ are countable, or at least one of them is co-countable, i.e. its complement is countable. You can now use the fact that "a countable union of countable sets is countable" to show that either $A$ or $A^c$ will be countable.