Discussion
A metric space is compact if and only if it is complete and :
Answer: B
A metric space is compact if and only if it is complete and totally bounded.
Total boundedness means that for every \(\epsilon \gt 0\) the space is covered by finitely many \(\epsilon\)-balls. With completeness, it gives sequential compactness: from any sequence one extracts a Cauchy subsequence by repeated pigeonholing into finitely many small balls, and completeness makes it converge.
Boundedness is not enough: the closed unit ball of \(l_{2}\), or \(\mathbf{R}\) with the metric \(\min(1, |x - y|)\), is complete and bounded but not compact. 'Closed' adds nothing to a complete space, and countability is unrelated (\([0, 1]\) is compact and uncountable).
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