Define d on the set {cat, dog, fish} by d(x, x) = 0 and d(x, y) = 1 whenever x ≠ y. Is this a valid metric?
ANo, because distance requires a notion of subtraction, which words do not have
BNo, because d(x, y) must depend continuously on x and y
CYes — it satisfies non-negativity, symmetry, and the triangle inequality (since 1 ≤ 1 + 1)
DYes, but only if the set has a vector space structure, which this set lacks
This is the discrete metric, and it is a perfectly valid metric on any set — including abstract sets of words. Non-negativity: d ≥ 0, and d(x, x) = 0. Symmetry: d(x, y) = d(y, x) by definition. Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z) becomes 1 ≤ 1 + 1 = 2 when x ≠ z, which holds. This example illustrates that a metric is purely axiomatic — it requires no algebraic structure on the underlying set, no continuity, and no connection to subtraction.
Question 2 Multiple Choice
In the taxicab metric on ℝ² defined by d((x₁, y₁), (x₂, y₂)) = |x₁ − x₂| + |y₁ − y₂|, what does the open ball B((0,0), 1) look like geometrically?
AA circle of radius 1 centered at the origin, identical to the Euclidean ball
BA square with sides parallel to the axes, with corners at (±1, 0) and (0, ±1)
CA square rotated 45°, with vertices at (±1, 0) and (0, ±1)
DThe entire plane, since taxicab distance is always less than Euclidean distance
The taxicab ball B((0,0), 1) = {(x, y) : |x| + |y| < 1}. This is a square rotated 45° — a diamond shape — with vertices at (1,0), (0,1), (−1,0), (0,−1). Option B describes a different square (with sides parallel to axes). Option A is the Euclidean ball — a circle. This geometric difference illustrates how the same underlying set (ℝ²) with different metrics generates different open balls and potentially different topologies.
Question 3 True / False
The same underlying set can carry multiple different metrics, which may produce different notions of 'closeness' and even different topologies.
TTrue
FFalse
Answer: True
This is a central insight: a metric is structure *imposed on* a set, not inherent to it. The Euclidean metric, taxicab metric, and discrete metric on ℝ² are all valid metrics, but they produce different open balls and different families of open sets. The Euclidean and taxicab metrics on ℝ² are actually topologically equivalent (they generate the same open sets), but the discrete metric generates a strictly different topology — every subset is open. The same set, different structures.
Question 4 True / False
Most metric on a vector space is equivalent to the metric induced by some norm on that space.
TTrue
FFalse
Answer: False
Metrics are more general than norms. A norm requires a vector space and satisfies homogeneity (‖λv‖ = |λ|‖v‖) and the triangle inequality; every norm induces a metric via d(x, y) = ‖x − y‖. But you can put metrics on vector spaces that no norm induces — for instance, the discrete metric on ℝ (d = 0 or 1) cannot come from any norm, since norms scale with scalar multiplication. Metrics apply to any set; norms require vector space structure. The Common Misconceptions section flags this directly.
Question 5 Short Answer
Why is the triangle inequality the most essential of the three metric axioms? What would fail about the concept of 'distance' if it were dropped?
Think about your answer, then reveal below.
Model answer: The triangle inequality d(x, z) ≤ d(x, y) + d(y, z) expresses that the direct route between two points is never longer than going via a detour. Without it, 'closeness' becomes incoherent: you could have x very close to y and y very close to z, yet x and z arbitrarily far apart. This would mean that 'being near' is not transitive in any useful sense, and the topological notion of a limit — where points within ε of a center form a coherent neighborhood — would break down. The triangle inequality is what makes the open ball B(x, r) a sensible notion of 'all points near x.'
Non-negativity and symmetry are relatively weak requirements. The triangle inequality is the load-bearing axiom that gives metric spaces their geometric and topological character. It ensures that open balls overlap in controlled ways, that sequences can converge meaningfully, and that the induced topology has the Hausdorff property. Removing it produces a structure that fails to behave like distance in any intuitive or mathematically useful sense.