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

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    Sets don't have either the property of completeness or compactness. Compactness is a property of topological spaces (http://en.wikipedia.org/wiki/Topological_space) and completeness is a property of metric spaces (http://en.wikipedia.org/wiki/Metric_space) or, if you prefer, uniform spaces (http://en.wikipedia.org/wiki/Uniform_space).2012-10-08
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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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