Hello All,
I am aware that this question is probably very simple (as I have scrolled around looking for similar questions) and therefore you can probably tell I am new to set theory and real analysis. So up front I am looking to understand a homework problem I did not get - to find:
1.) an infinite collection of subsets of $\mathbb{R}$ that contains $\mathbb{R}$ 2.) is closed under formation of countable unions 3.) is closed under formation of countable intersections 4.) is NOT a $\sigma$-algebra
So seeing this I immediately thought of a set that did not support closure under complementation, as this is the only remaining requirement in the list of a $\sigma$-algebra. I am quite lacking in knowledge about countability and the standard number systems, and am aware of the "standard" example utilizing the positive integers where the subsets are finite or their complements are finite - I understand that. The problem is that this is on the reals, which is uncountable (not countably infinite) and the only poor example I can think of is $\mathbb{R} - \varnothing$.
Any help would be greatly appreciated.
Brett
