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TIFR Mathematics 2022 | Part A | Question-1

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

  1. $(X, d)$ is complete as a metric space.
  2. 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?

  1. (I) implies (II) and (II) implies (I).
  2. (I) implies (II) but (II) does not imply (I).
  3. (I) does not imply (II) but (II) implies (I).
  4. (I) does not imply (II) and (II) does not imply (I).
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