How Many Are Closed Sets?

Consider the metric space $\mathbb{R}^2$ equipped with the standard Euclidean distance

$d\big((x_1, x_2), (y_1, y_2)\big) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}.$

How many of the following subsets $S \subset \mathbb{R}^2$ are closed in this metric space?

• $S = \{(x,y) \, : \, x^2 +y^2 = 1\}$
• $S = \{(x,y) \, : \, x^2 +y^2 \le 1\}$
• $S = \{(x,y) \, : \, x \in \mathbb{Q}, y \in \mathbb{Q} \}$
• $S = \{(x,0) \, : \, x\in \mathcal{C}\}$, where $\mathcal{C} \subset \mathbb{R}$ is the middle-thirds Cantor set