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Determine whether the given subset in $\mathbb{R}^2$ with the Euclidean metric is closed and whether it is complete?

  1. $S^1=\{(x_1,x_2) \in \mathbb{R}^2\mid x_1^2 + x_2^2 =1\}$;

  2. $B_1=\{(x_1,x_2) \in\mathbb{R}^2\mid x_1^2 + x_2^2 <1\}$;

  3. $B_1^c=\{(x_1,x_2)\in \mathbb{R}^2\mid x_1^2 + x_2^2 \geq1\}$

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    The pipe `|` is not a delimiter so the LaTeX does not apply spacing, using `\mid` however works nicer.2012-03-01

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