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 ??

Well, (1) and (2) are obvious.
Hint for (3): let $A_1,A_2,\dots\in \mathcal A$. Consider two cases:


*

*Either all $A_n$'s contain only countable points.

*Or at least one of them (w.l.o.g., say $A_1$) contains all but countable points.

