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](https://math.stackexchange.com/questions/114170/infinite-closed-partition-of-the-real-numbers-with-a-certain-property). While an interesting construction of a partition with no closed _uncountable_ unions was given, there has not yet been a conclusive answer.

Two observations:
<ol>
<li>No infinite subcollection of the partition can be locally finite since the union of a locally finite collection of closed sets is closed.</li>
<li>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.
</ol>