Infinite sigma-algebra

[Zaare]

I'm supposed to answer the question "Can a sigma-algebra be infinite and countable?"
I think I can show that if it has a countable number of disjoint subsets, then it can't be countable considering the possible combinations of the subsets.
Now I need to show that if a sigma-algebra consists of an infinite number of subsets, then it has a countable number of disjoint subsets.
Any ideas on how I can do this?

[chingkui]

To the original question:
http://planetmath.org/?op=getmsg&id=5848

[Hurkyl]

I think I can show that if it has a countable number of disjoint subsets, then it can't be countable considering the possible combinations of the subsets.

No you can't.

Now, if you instead said countably infinite... :smile:


Now I need to show that if a sigma-algebra consists of an infinite number of subsets, then it has a [countably infinite] number of disjoint subsets.

(I edited it)

Proof by contradiction, maybe?

[Zaare]

No you can't.

Now, if you instead said countably infinite... :smile:


That's what I meant. I was sloppy. :redface:
Thank you both for the help.