Dear mathstack community,
I have been struggling to understand the definition of the atomless $\sigma$ algebra consisting of the collection of periodic subsets of $ω1$. Defined here (in the answer of Joel Hamkins) :
https://mathoverflow.net/questions/22477/sigma-algebra-without-atoms
In particular, I had the following questions:
(1) How does complementation work ?
(2) Could you give me a concrete example of how the join of two elements is formed in this algebra?
(3) Could you give me a concrete example of how the meet of two elements is formed in this algebra?
Also if you could simply point out some standard text book where I can read more about this particular example if would be very grateful (The wikipedia entrance I did not find very helpful). Thanks in advance!
(1) $\neg A=\omega_1\setminus A$. To see that this is a periodic set note that to build a set $A$ in the $\sigma$ algebra you start with a countable ordinal $\alpha$ and some set $a\subseteq \alpha$, and then "translate $a$" until you have a periodic set. Then $\omega_1\setminus A$ is obtained starting with the same $\alpha$, but instead taking $\alpha\setminus a$ as the subset to repeat.
(2) by taking their union.
(3) by taking their intersections.
Unfortunately I don't know a reference discussing this example in particular.