Question regarding the finite intersection property Suppose I have a sequence $\{C _j \} $ of closed and bounded intervals such that $ \cap _{j=1 } ^{\infty } C _j= \emptyset$. Does it follow that for some $n $, we must have $\cap _{j } ^n E _j = \emptyset$
I believe this is the only possibility since otherwise the assumption would violate the finite intersection property.

It  can be concluded iff the space is compact .If the space is compact then for every sequence of closed intervals possessing finite intersection property ,the intersection of the entire collection is non empty
