For (1), how about the collection ?
There is a theorem that states,
If there is a collection of compact subsets of a metric space which statifies the Finite Intersection Property, then .
I am trying to find examples of collections of subsets from the reals,
where the words
replace the word "compact" in the above statement, and show that the theorem is not true for either closed or bounded subsets, so the theorem only holds for compact subsets.
That would imply, there exists collections of (1) closed or (2) bounded subsets whose intersection IS the null set.
Thanks! Hope the question is clear... I am seeking two counterexamples, one each for collections of closed subsets and collections of bounded subsets.