For a system of sets closed under union and complement, does there always exists a partition generating it Let $\mathcal E \subseteq \mathcal P(X)$ be a non-empty system of sets over $X$ which is closed under arbitrary union and complement (and hence also by intersection and set difference).
Then is it always possible to find a partition of $X$ in $\mathcal E$, i.e. a family of sets $\{ P_i \}_{i\in I}$, $P_i \in \mathcal E$, whose union is $X$ and which are pairwise disjoint, such that every $A \in \mathcal E$ could be written as a union of sets from $\{ P_i \}$?
If $X \subseteq \mathbb N$, then I am able to prove this claim. But for arbitrary $X$ I am not able to proof it, but I am also unable to find a counter example. I conjecture it to be true even in the general setting, as the closure properties are very unrestrictive, but if it is true; how to prove it, or otherwise do you know a counterexample?
 A: The statement is true: let $\sim_\mathcal{E}$ be defined by $$x\sim_\mathcal{E}y\iff \forall A\in\mathcal{E}(x\in A\iff y\in A).$$ Then the $\sim_\mathcal{E}$-equivalence classes partition $X$, and induce $\mathcal{E}$, since $A\in\mathcal{E}=\bigcup_{x\in A} [x]_{\sim_\mathcal{E}}$. Moreover, since $$[x]_{\sim\mathcal{E}}=\bigcap_{x\in A\in\mathcal{E}}A,$$ and $\mathcal{E}$ is closed under arbitrary intersections, each $[x]_{\sim_\mathcal{E}}$ is in $\mathcal{E}$.

If, however, we demand only closure under finite unions/intersections, the answer is no: defining $\sim_\mathcal{E}$ as before, it's easy to show that $\mathcal{E}$ is generated by a partition iff it is generated by the partition in $\sim_\mathcal{E}$-classes. But let $X$ be any totally disconnected topological space in which no singleton is clopen (e.g. Cantor space), and let $\mathcal{E}$ be the collection of clopen sets in $X$. Then:


*

*$\mathcal{E}$ is closed under finite intersections and under complements

*$[x]_{\sim_\mathcal{E}}=\{x\}$, but

*no singleton is clopen.
