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Consider the following properties of a metric space $(X, d)$ :

- $(X, d)$ is complete as a metric space.
- For any sequence $\left\{Z_{n}\right\}_{n \in \mathbb{N}}$ of closed nonempty subsets of $X$, such that $Z_{1} \supseteq Z_{2} \supseteq$ $\ldots$ and \[ \lim _{n \rightarrow \infty}\left(\sup _{x, y \in Z_{n}} d(x, y)\right)=0, \] $\bigcap_{n=1}^{\infty} Z_{n}$ is a singleton set.

Which of the following sentences is true?

- (I) implies (II) and (II) implies (I).
- (I) implies (II) but (II) does not imply (I).
- (I) does not imply (II) but (II) implies (I).
- (I) does not imply (II) and (II) does not imply (I).