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I want to find an example of set which is complete but not compact.

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    Possible duplicate of [Example of a complete metric space which is not compact](http://math.stackexchange.com/questions/1049266/example-of-a-complete-metric-space-which-is-not-compact). It was asked a year later, but it was better worded. No +7 answer, but basically the same counterexample.2016-04-27

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$\mathbb{R}$ is complete but not compact.

In fact, a uniform space is compact iff it is complete and pre-compact. So in a finite dimensional normed space, every nonbounded closed subset works.

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take a unit circle in complex then apply the hyperbolic metric the set is complete but not compact hyperbolic metric(ds2)=dx2+dy2/(1-x2-y2)2

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    Please add some more detail.2014-04-03