Define d: ℝ × ℝ → ℝ by d(x, y) = (x − y)². Is d a valid metric on ℝ?
AYes — it is non-negative, zero only when x = y, and symmetric
BNo — it fails symmetry: d(x, y) ≠ d(y, x) in general
CNo — it fails the triangle inequality: d(x, z) can exceed d(x, y) + d(y, z)
DNo — it fails non-negativity: squared values can be negative
d(x, y) = (x−y)² satisfies non-negativity, the identity condition, and symmetry. However, it fails the triangle inequality. For example: d(0, 2) = 4, but d(0, 1) + d(1, 2) = 1 + 1 = 2 < 4. A detour through 1 is 'shorter' than going directly from 0 to 2 under this function. The triangle inequality is the most substantive of the three axioms — squaring destroys the additive structure it requires.
Question 2 Multiple Choice
In the discrete metric on a set X (where d(x, y) = 1 for x ≠ y and d(x, x) = 0), what does the open ball B(p, r) look like?
AFor any r > 0, the open ball contains all points within Euclidean distance r of p
BFor r ≤ 1, B(p, r) = {p}; for r > 1, B(p, r) = X
CThe open ball always contains exactly the two nearest neighbors of p
DOpen balls are not defined in the discrete metric because all points are equidistant
In the discrete metric every pair of distinct points is distance 1 apart. B(p, r) = {x : d(x, p) < r}. If r ≤ 1, only p itself satisfies d(p, p) = 0 < r, so the ball is just {p}. If r > 1, then d(q, p) = 1 < r for every q ≠ p, so the ball is all of X. This means every singleton {p} is open (take r = 1/2), and therefore every subset of X is open — a radically different topology from the real line.
Question 3 True / False
The same underlying set can be equipped with different metrics, producing different notions of convergence and different collections of open sets.
TTrue
FFalse
Answer: True
This is one of the central points of the metric space abstraction. On the set of continuous functions on [0, 1], the supremum metric d(f, g) = sup|f(x) − g(x)| defines uniform convergence, while the L² metric defines mean-square convergence. A sequence can converge in one metric but not the other. The metric determines the entire analytical character of the space — the set alone determines none of this.
Question 4 True / False
The function d(x, y) = |x − y|² is a valid metric on ℝ.
TTrue
FFalse
Answer: False
Despite satisfying non-negativity, identity, and symmetry, d(x, y) = (x − y)² fails the triangle inequality. For x = 0, y = 1, z = 2: d(0, 2) = 4, but d(0, 1) + d(1, 2) = 1 + 1 = 2. Since 4 > 2, the inequality d(x, z) ≤ d(x, y) + d(y, z) is violated. The standard metric d(x, y) = |x − y| is valid; raising to any power greater than 1 typically destroys the triangle inequality on ℝ.
Question 5 Short Answer
Why is the triangle inequality the most mathematically significant of the three metric axioms, and what would break down if it failed?
Think about your answer, then reveal below.
Model answer: The triangle inequality ensures that direct paths are never worse than detours, and that nearness is transitive in the right sense. Without it, a point could be far from p yet very close to something very close to p — making limits, continuity, and Cauchy sequences incoherent. It is what allows open balls to overlap sensibly and enables the entire machinery of analysis.
Non-negativity and symmetry capture basic definitional intuitions about distance. The triangle inequality is the substantive constraint that makes metric geometry work. You can compose paths, estimate distances indirectly, and prove that Cauchy sequences are eventually close to each other — all rely on it. When verifying a proposed metric, the triangle inequality is always the hard and interesting step.