Beautiful conterexample.
I was trying to come up with a conterexample in using the definition I posed above (same idea) but I used finite number of sets. And I forgot the important rule of analysis (or topology here): what works for finite sets does not necessarily work for infinite sets.
So I am guessing if was such that then the above would actually be true. Correct? Maybe this is consequence of Heine-Borel theorem, no?
The nature of a standard metric topology makes it impossible to find such an example. However, I quite sure that one could construct a general (probability a finite) topological space where that property is true. I just not have thought about it in a very long time. Good Luck!