I was reading this Wiki article: http://en.wikipedia.org/wiki/Sigma-algebra and don't quite understand one of the examples.
"The collection of subsets of X which are countable or whose complements are countable (which is distinct from the power set of X if and only if X is uncountable.). This is the σ-algebra generated by the singletons of X."
1. Σ is not empty,
2. Σ is closed under complements: If E is in Σ then so is the complement (X \ E) of E,
3. Σ is closed under countable unions: The union of countably many sets in Σ is also in Σ.
The Attempt at a Solution
I kind of understand sigma-algebra, but I really don't get this example... If it's the sigma-algebra generated by singletons, then how can the first property be satisfied?