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Let $d(x, y)$ be the usual Euclidean metric on $\mathbb{R}^{2}$. Which of the following metric spaces is complete?

  1. $\mathbb{Q}^{2}\subset\mathbb{R}^{2}$ with the metric $d(x, y)$.
  2. $[0, 1]\times [0, \infty)\subset \mathbb{R}^{2}$ with the metric $d'(x, y)=\frac{d(x, y)}{1+d(x, y)}$.
  3. $(0, \infty)\times [0, \infty) \subset \mathbb{R}^{2}$ with the metric $d(x, y)$.
  4. $[0, 1]\times [0, 1) \subset \mathbb{R}^{2}$ with the metric $d''(x, y) = \min \left \{ 1, d(x, y) \right \}$.
asked in Linear Algebra by Boss (29.5k points) | 165 views

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