Counterexample

Is this true?

Bounded ⇒ totally bounded

No — it is false.

The counterexample

$\mathbb{R}$ with the discrete metric

Why it fails

Everything is within distance 1, but no finite set of balls of radius ½ covers it.

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Keep practicing

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

See real exam options built on this mistake Drill statements like this

compactness metric spaces

Go deeper· Explained in full in Compactness — the implication map· Asked in CSIR NET June 2025· From Metric Spaces › Compactness: open covers, sequential, Heine–Borel

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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