Metric spaces & topology of the line

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   In $\mathbb{R}^{2}$ with the Euclidean metric $d_{2}(p, q) = \sqrt{(p_{1} - q_{1})^{2} + (p_{2} - q_{2})^{2}}$, compute $d_{2}(P, Q)$ in exact (simplified radical) form for $P = (-3, 3)$ and $Q = (2, -4)$.

2. Warm-up

   In $\mathbb{R}^{2}$ with the Euclidean metric $d_{2}(p, q) = \sqrt{(p_{1} - q_{1})^{2} + (p_{2} - q_{2})^{2}}$, compute $d_{2}(P, Q)$ in exact (simplified radical) form for $P = (-1, 3)$ and $Q = (1, 4)$.

3. Warm-up

   In $\mathbb{R}^{2}$ with the Euclidean metric $d_{2}(p, q) = \sqrt{(p_{1} - q_{1})^{2} + (p_{2} - q_{2})^{2}}$, compute $d_{2}(P, Q)$ in exact (simplified radical) form for $P = (1, -2)$ and $Q = (2, -3)$.

Problems shown under their original terms: Generated (template, solver-computed).