2.II.14G

Analysis II | Part IB, 2004

Let XX 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 X,A1A2X, A_{1} \supset A_{2} \supset \cdots, can be empty. Show that if we also assume that

limndiam(An)=0\lim _{n \rightarrow \infty} \operatorname{diam}\left(A_{n}\right)=0

then the intersection is not empty. Here the diameter diam(A)\operatorname{diam}(A) is defined as the supremum of the distances between any two points of a set AA.

We say that a subset AA of XX is dense if it has nonempty intersection with every nonempty open subset of XX. Let U1,U2,U_{1}, U_{2}, \ldots be any sequence of dense open subsets of XX. Show that the intersection n=1Un\bigcap_{n=1}^{\infty} U_{n} is not empty.

[Hint: Look for a descending sequence of subsets A1A2A_{1} \supset A_{2} \supset \cdots, with AiUiA_{i} \subset U_{i}, such that the previous part of this problem applies.]

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