Let $Y$ be a collection of subsets of the set X. Show that for each $A \in \sigma(Y)$ there is a countable subfamily $B_0 \subset Y$ such that $A\in \sigma(B_0)$
My try: I look at $\cup B_i$ where $B_i$ is a countable subfamily of $Y$. And I want to show that $Y\subset \cup B_i \subset\sigma(Y)$. Both the $\cup B_i \subset\sigma(Y)$ and $Y\subset \cup B_i$ feels intuitive, but how do I write it out rigorously?