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
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