Is it correct to say that $ \left \{ \left \{ x \right \}, s.t. 0\leq x\leq 1 \right \}$ is a partition of $[0;1]$. (Using this partition definition: https://en.wikipedia.org/wiki/Partition_of_a_set#Definition_and_Notation ) And that: $ B([0;1])=\sigma (\left \{ x \right \}, s.t. 0\leq x\leq 1) $.
More over in this case can we writte: $ \forall F \in B([0;1]) \exists {\alpha_{k \in \mathbb{R}}} \: s.t. \: \forall k \: , \alpha_k \in \: \left \{ 0;1 \right \}$ such that $ F=\bigcup_{k \in [0;1]}^{} \alpha_k (\left \{ x \right \}) $
That means that $ \forall F \in B([0;1]) $ , F can be describe by an union, countable or not, of element from the partition $\left \{ \left \{ x \right \}, s.t. 0\leq x\leq 1 \right \}$ of $B([0;1])$?
And more generaly does for every set $\sigma(S)$ it exists a partition $P$ of $S$ such that $\forall F \in\sigma(S)$ can be describe by an union of element from $P$?