We know that a closed subset of a complete metric space is complete. But I want to find a closed subset $A$ of an incomplete metric space $(X,d)$ such that $A$ is not complete.
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9Why not take $A=X$? – Kenny Wong Jul 24 '20 at 18:15
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The set $\mathbb{Q}$ is a metric space that isn't complete. The set $[0,\infty)\cap\mathbb{Q}$ is closed in $\mathbb{Q}$ and is not complete. – Josh B. Jul 24 '20 at 18:16
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1A simple example is $X=\mathbb{R}\setminus{0}$, $A=\mathbb{R}^+\setminus{0}$. – Chrystomath Jul 24 '20 at 18:21
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Yes, I got it. Thanks – user811319 Jul 25 '20 at 02:55
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Let $x$ be an irrational number and use the interval $(-\infty,x] \cap \mathbf Q$, which is the same as $(-\infty,x) \cap \mathbf Q$ since $x$ is irrational.
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Let $(\mathbf Q,|\cdot|)$ be a metric space which is incomplete. Consider $\mathbf Q$ to be a subspace of $\mathbf Q$.
$\mathbf Q$ is closed in $\mathbf Q$, and also incomplete.
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