A student proposes d(x, y) = (x − y)² as a metric on ℝ. She notes it is always nonnegative, equals zero only when x = y, and is symmetric. Is this a valid metric?
AYes — all three metric axioms are satisfied
BNo — it fails positivity because (x − y)² can be arbitrarily large
CNo — it fails the triangle inequality: d(0, 2) = 4, but d(0, 1) + d(1, 2) = 1 + 1 = 2 < 4
DNo — squaring is not a symmetric operation, so the symmetry axiom fails
Positivity and symmetry both hold, but the triangle inequality fails. d(0, 2) = 4, while d(0, 1) + d(1, 2) = 1 + 1 = 2. Since 4 > 2, the requirement d(x, z) ≤ d(x, y) + d(y, z) is violated. Passing two out of three axioms is not enough — all three must hold. The triangle inequality is often the hardest to verify and the most commonly violated by plausible-seeming distance functions.
Question 2 Multiple Choice
The set of continuous functions on [0,1] with d(f, g) = sup|f(x) − g(x)| forms a metric space. What does this example most directly illustrate?
AMetric spaces only work for finite-dimensional real vector spaces like ℝⁿ
BThe supremum norm is always the most natural metric for function spaces
CThe metric axioms are flexible enough to apply to sets whose 'points' are functions, not just numbers
DAny set equipped with a supremum operation automatically satisfies the metric axioms
The 'points' in this metric space are functions — infinite-dimensional objects. d(f, g) measures the worst-case gap between two functions. All three axioms hold: d(f, g) ≥ 0, it equals 0 iff f = g everywhere, it is symmetric, and the triangle inequality holds for suprema. The key message is that the three axioms are an abstract template that applies to any reasonable notion of distance, regardless of what the 'points' are — numbers, vectors, functions, or anything else.
Question 3 True / False
Every metric space is automatically a topological space.
TTrue
FFalse
Answer: True
A metric generates a topology: the open balls B(x, r) = {y : d(x, y) < r} form a basis for a topology on X, and open sets are unions of open balls. This metric topology satisfies all topological axioms. The converse is false — not every topological space has a metric that generates it (non-metrizable spaces exist). Metric spaces are topological spaces with extra structure (an explicit distance function), which is why studying them first gives concrete intuition for the more abstract topological setting.
Question 4 True / False
The triangle inequality is the least important metric axiom because it merely captures an obvious geometric fact about straight-line distances.
TTrue
FFalse
Answer: False
The triangle inequality is arguably the most important axiom. It makes 'closeness' transitive: if x is close to y and y is close to z, then x must be reasonably close to z. This transitivity is essential for convergence — without it, you couldn't chain together bounds or conclude that a sequence approaching a limit stays near it. Positivity and symmetry are sanity conditions; the triangle inequality is the structural condition that makes the distance function useful for proving theorems in analysis.
Question 5 Short Answer
Why is the triangle inequality the 'load-bearing' axiom of the metric space definition — what breaks down if you remove it?
Think about your answer, then reveal below.
Model answer: The triangle inequality makes closeness transitive: if d(x, y) < ε and d(y, z) < ε, then d(x, z) < 2ε. Without it, two points each 'close' to a third need not be close to each other, which breaks the concept of convergence. A sequence could have each term close to the limit but terms arbitrarily far from each other, destroying Cauchy-type arguments. Most bounding arguments in analysis chain inequalities of the form d(a, c) ≤ d(a, b) + d(b, c); this structure depends entirely on the triangle inequality.
Positivity says distances are nonnegative; symmetry says direction doesn't matter. These are sanity checks. The triangle inequality is the condition that makes the distance function structurally useful — it ensures open balls behave like neighborhoods, that limits are unique, and that convergence is a meaningful concept worth studying.