I'm trying to understand a proof of the fact that the sigma algebra generated by open intervals(lets call this A) is equivalent to the sigma algebra generated by half open intervals(lets call this B). The gist of the proof seems to be to show that a half open set can be written as an intersection of open intervals and an open interval can be written as a union of half open intervals. These two facts are used to show that each is the subset of the other. My trouble with this is how do we know that when we take an arbitrary set in A, that it is in fact an open interval or that when we take an arbitrary set in B it is in fact a half open interval. There are so many other sets in A and B that seem to be unaccounted for, so I dont quite understand how we can know from those two facts alone that each is a subset of the other. I'd appreciate any help that can be offered. Thanks very much.

If , then .

thanks, that works.