Describe the sigma algebra generated by singleton subsets Let's denote the set of all singleton subsets of $X$(i.e. of all subsets consisting of one element) by $A$. Describe $\sigma(A)$ in the following two cases:
i) $X$ is countable
ii) $X$ is uncountable
I am new to this topic, so could you please help me understand the thinking/the concept behind this problem..
 A: We want to find the smallest $\sigma$-algebra that contains $A$.  In the first case, for any subset $Y$ of $X$, we can express $Y$ as a countable union of singletons in $X$, so $Y \in \sigma(A)$.  But $Y$ was arbitrary, so every subset of $X$ is in $\sigma(A)$.
The second case is a bit tougher.  We know that $\sigma(A)$ will contain all of the countable subsets of $X$, but it will also contain their complements, which aren't going to be countable.  Let's call a set co-countable if its complement is countable.
Try showing that the collection of countable subsets and co-countable subsets of $X$ is a $\sigma$-algebra.  What's its relationship with $\sigma(A)$?
A: First of all, a remark: a countable set can be written as a countable union of singletons, and $\sigma$-algebra are stable under countable unions, hence in each case, $\sigma(\mathcal A)$ contains each countable subset of $X$.
In the first case, we are done. 
In the second one, we also have to consider complements of such sets. It remains to deal with the following question: are there other ones?
