3
$\begingroup$

The maximal sigma algebra on a set is its power set.

When the set is countable, its maximal sigma algebra can be generated by all singleton subsets, i.e. subsets each consisting of exactly one element.

Conversely, if the maximal sigma algebra on a set can be generated by all singleton subsets, must the set be countable?

Thanks and regards!

1 Answers 1

15

Assume $X$ is uncountable. The countable-cocountable $\sigma$-algebra, that is to say the $\sigma$-algebra consisting of the sets which are countable or have countable complement, contains the singletons, so the $\sigma$-algebra generated by the singletons can't be the entire power set (note that $X$ contains two disjoint uncountable sets).

It is an easy exercise to prove that the singletons generate the countable-cocountable $\sigma$-algebra on any set. If the set happens to be countable then the countable-cocountable $\sigma$-algebra coincides with the power set.


Edit: For people interested in choice-specific issues, our resident connoisseur Asaf recommends perusal of the following links:

  • 3
    Please no more choice comments to this answer this is veering too far off-topic!2012-01-09