A union of finite sets can always be converted to a union of finite disjoint sets.
I was wondering if a union of countable sets can always be converted to a union of countable disjoint sets? If yes how to do that?
Thanks!
Update with Paul's question:
"I wonder about the natural extension of this question to uncountable unions. Can we always refine an arbitrary set {Aα} to {Bα} (i.e. with Bα⊂Aα for each α) such that ⋃αBα=⋃αAα? The answer below uses countability pretty blatantly and I can't think of a way to do it without. "
Thanks!