How Many Are Closed Sets?

Probability Level 3

Consider the metric space R2\mathbb{R}^2 equipped with the standard Euclidean distance

d((x1,x2),(y1,y2))=(x1y1)2+(x2y2)2.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 SR2S \subset \mathbb{R}^2 are closed in this metric space?

  • S={(x,y):x2+y2=1}S = \{(x,y) \, : \, x^2 +y^2 = 1\}
  • S={(x,y):x2+y21}S = \{(x,y) \, : \, x^2 +y^2 \le 1\}
  • S={(x,y):xQ,yQ}S = \{(x,y) \, : \, x \in \mathbb{Q}, y \in \mathbb{Q} \}
  • S={(x,0):xC}S = \{(x,0) \, : \, x\in \mathcal{C}\}, where CR\mathcal{C} \subset \mathbb{R} is the middle-thirds Cantor set
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