2.II.14G
Let be a non-empty complete metric space. Give an example to show that the intersection of a descending sequence of non-empty closed subsets of , can be empty. Show that if we also assume that
then the intersection is not empty. Here the diameter is defined as the supremum of the distances between any two points of a set .
We say that a subset of is dense if it has nonempty intersection with every nonempty open subset of . Let be any sequence of dense open subsets of . Show that the intersection is not empty.
[Hint: Look for a descending sequence of subsets , with , such that the previous part of this problem applies.]
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