Why is the set of all intervals on [0,1] not a sigma algebra I am working my way through "A First Look at Rigorous Probability Theory" by Rosenthal and I'm a bit confused by something. In chapter 2 p.9 he mentions the set $J$ as being all intervals (open/closed/half-open/singleton/empty) contained in $[0,1]$ He then goes on to say that $J$ is not a $\sigma$-algebra. My question is why not? I have gathered by trawling this site that it fails the condition because it is not closed under countable unions - could someone please provide a sort of "proof" of that?

The set $J$ generates a $\sigma$-algebra, but $J$ itself is not one, since for example, the complement of an open interval is not an interval.