Non-Separable Sigma-Algebras: Examples & Properties

[wayneckm]

Hello all,

May someone give me an example of sigma-algebra which is not countably generated?

Apparently such example can only be found in a non-separable space?

Taking $$\mathbb R$$ as example,

1) Sigma-algebra generated by any subsets of a separable space is countably generated?

2) That in a non-compact space may also be countably generated?

3) That in a compact space is countably generated?

Please kindly address my correctness of the above statements. Thanks very much.

Wayne

 

[Valkarie]

1) That is correct - any sigma-algebra generated by any subset of a separable space is countably generated. 2) That is not necessarily true. For example, consider the Borel sets in a non-compact space such as [0,1] with the usual Euclidean topology. The Borel sets are not countably generated.3) That is correct - any sigma-algebra generated by a compact space is countably generated.