Discussion
Which of the following statements is not true ?
Answer: A
The false statement is 'Every closed and bounded subset of a metric space is compact'.
Closed and bounded implies compact in \(\mathbf{R}^{n}\) (Heine-Borel), but not in a general metric space. Take \(\mathbf{R}\) with the discrete metric: the whole space is closed and bounded (every distance is at most 1), yet the cover by singletons has no finite subcover. Likewise the closed unit ball of an infinite-dimensional Hilbert space is closed and bounded but not compact.
The other three are true. A compact subset of a metric space is closed (metric spaces are Hausdorff) and bounded (cover it by balls of radius 1). A closed subset of a compact set is compact. A Cartesian product of compact sets is compact (Tychonoff, or directly for finitely many factors).
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