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Mar 18, 2012 at 9:11 history edited KP Hart CC BY-SA 3.0
Turned the partition into a family of closed sets.
Mar 18, 2012 at 9:04 comment added KP Hart Indeed, I'll edit the answer to reflect your suggestion.
Mar 18, 2012 at 3:06 vote accept LostInMath
Mar 18, 2012 at 3:05 comment added LostInMath After a bit of ingenuity to consider the space of closed subsets it becomes so easy! Thank you very much. By the way, if I understand it correctly, one should perhaps add $0$ and $1$ to each element of $\mathcal{F}$ to make sure they are closed in $[0,1]$ too.
Mar 16, 2012 at 23:12 history answered KP Hart CC BY-SA 3.0