Suppose that C is a collection of closed subsets of the real

Suppose that C is a collection of closed subsets of the real line with the finite intersection property. Is the intersection of all sets in C necessarily nonempty? What if at least one of the sets in C is compact?

Solution

Recall that

A metric space X is compact if and only if every collection C of closed sets in X with the finite intersection property has a nonempty intersection.

Here, X = R, a metric space and so, it has nonempty intersection. Thus, each collection must be non-empty.

If atleast one of the sets of collection C is compact, then by Compactness of metric spaces Theorem, X must be sequentially compact and so it must be totally bounded and complete.

Hope this helps!