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

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up vote 9 down vote accepted

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 '14 at 17:12

Just to throw one more answer into the mix, any infinite set with the discrete metric is bounded but not totally bounded (I like @Gedgar's answer better, but just for some variety...)

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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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Every totally bounded set is bounded. But not conversely. The unit ball in Hilbert space is bounded, but not totally bounded.

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Maybe you mean an infinite dimensional Hilbert space? – Cameron Williams Mar 22 '14 at 17:12

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