Is there a partition of the real line into infinitely many closed subsets so that no infinite union of these subsets (except the whole space) is closed?
This question was asked also at math.stackexchange.com. While an interesting construction of a partition with no closed uncountable unions was given, there has not yet been a conclusive answer.
- No infinite subcollection of the partition can be locally finite since the union of a locally finite collection of closed sets is closed.
- The partition must be uncountable since the real line cannot be partitioned into countably many (and $\geq 2$) closed subsets by a theorem of Sierpiński.