# Is there any difference between Bounded and Totally bounded?

Is there any difference between bounded and totally bounded? (in a metric space)

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The real line $\Bbb R$ endowed with the metric $d(x,y)=\min(1,|x-y|)$ is a bounded metric space that isn't totally bounded.

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+1: Nice, simple example. –  copper.hat Mar 22 at 17:12

A major theorem in metric space theory is that a metric space is compact if and only if it is complete and totally bounded. In $\mathbb{R}^{n}$ with the usual metric ( for $n < \infty$), bounded and totally bounded are the same, which is essentially the content of the Heine Borel theorem. In fact, the unit ball of a Banach space is compact if and only if the space is finite dimensional.

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