Determine whether the given subset in $\mathbb{R}^2$ with the Euclidean metric is closed and whether it is complete?
$S^1=\{(x_1,x_2) \in \mathbb{R}^2\mid x_1^2 + x_2^2 =1\}$;
$B_1=\{(x_1,x_2) \in\mathbb{R}^2\mid x_1^2 + x_2^2 <1\}$;
$B_1^c=\{(x_1,x_2)\in \mathbb{R}^2\mid x_1^2 + x_2^2 \geq1\}$