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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. – Asaf Karagila Mar 1 '12 at 0:38

$S^1$ and $B_1^c$ are closed and complete. Closure is easy, (the complements of each set are clearly open). The plane is complete. Since any Cauchy sequence in the plane converges, a Cauchy sequence in either of these two sets will converge and the limit is necessarily in the set by closure. However $B_1$ is open and not complete. Just take $\{1-1/n\}$. It converges to $1$ which doesn't belong to $B_1$.

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