if X not finite then O is not a $\sigma$ - algebra let O (family of sets) consist of those sets which are either finite or have a finite complement. then O is an algebra. I did this part! my question now is: if X (space) is not finite, then O is not a $\sigma$ - algebra. The definitions of algebra and $\sigma$ - algebra are clear to me. For the first part of the question it didn't matter whether X was finite, because if it wasn't then $X^{c}$ would, and therefore would be included in O. And since $\sigma$ - algebra differ from algebra in that we have closure under countable unions, I assume that I have to look there. But I am not sure how to go abut it. Any sorts of tip would be great, thank you very much.

_Hint:_ $X$ contains a countable subset with an infinite complement.