Due October 12
Let X be a metric space. We say that a collection of subsets of X has the finite intersection property (FIP) if any finite subcollection of them has nonempty intersection:
- X is compact if and only if the intersection of any collection of closed sets that has the FIP is nonempty.
- Give an example of a decreasing sequence of nonempty closed sets in a metric space with empty intersection.
- The closed ball is a closed set in X.
- Is in every metric space?
If is continuous, its graph is closed in .
If X is a metric space, then for any .