Why is this set a $\sigma$-algebra?? $X$ is an uncountable set.
Why is $\mathcal{A}=\{A \subset X: A \text{ or } X \setminus A \text{ is countable } \}$ a $\sigma$-algebra ??
$$$$
A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a collection of subsets of $X$ such that :
(1) $\varnothing \in \mathcal{A}$
(2) $A \in \mathcal{A} \Rightarrow X \setminus A \in \mathcal{A}$
(3) $A_n \in \mathcal{A} \Rightarrow \cup_{n=1}^{\infty} A_n \in \mathcal{A}$
$$$$
Could you give me a hint how to show that $\mathcal{A}=\{A \subset X: A \text{ or } X \setminus A \text{ is countable } \}$ is a $\sigma$-algebra ??

1 and 2 should be easy, for 3.
Given $A_n \in \mathcal{A}$, if each $A_n$ is countable, then we know $\cup_{n=1}^\infty A_n$ is countable by Andre's comment. 
Now suppose one of the $A_n$, say $A_{n_0}$ is uncountable, then $X\setminus A_{n_0}$ is countable, observe that 
$$(\cup_{n=1}^\infty A_n)^c =\cap_{n=1}^\infty A_n^c \subset A_{n_0}^c .$$
Any subset of a countable set is also countable.
