Can anyone help me out with an example of a closed set that is not complete? I have read up on the set of irrational numbers with the euclidean metric is such an example on other web pages, but that does not make any sense to me since the set of irrational numbers with the euclidean metric in not closed to begin with
2026-03-26 18:56:55.1774551415
Closed set that is not complete
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Yes, the set $\Bbb I$ of irrational numbers is an example. It is a closed subset of itself and it is not complete (with respect to the usual distance).
On the other hand, every closed subset of a complete metric space is always complete.