I am working on a homework problem and am somewhat lost. I know that an answer will not be given on a silver platter and am fine with that – I need to know what I am missing in understanding so that I can solve the problem.
I need an infinite collection of subsets of $\mathbb{R}$ that contains $\mathbb{R}$, is closed under the formation of countable unions and countable intersections, but is not a $\sigma$-algebra.
So I immediately thought that the only requirement not mentioned to make it a $\sigma$-algebra is the closure under complementation. That is why I thought of maybe using $\mathcal{P}(\mathbb{R})-\{\varnothing\}$, the powerset 'minus' the null set. Is this okay? Can you subtract 'nothing' like this? Otherwise I am quite lost and any direction would be greatly appreciated.
Nate
P.S> I could not find suitable suggestions to my question by looking around on the site.
Best Answer
Your proposed answer of $\mathcal{P}(\mathbb{R})-\{\varnothing\}$ is not closed under intersections, because $\{1\}\cap\{0\}=\varnothing$.
Try $\mathcal{P}(\mathbb{N})\cup\{\mathbb{R}\}$.