Alphabeta Math

Metric Spaces: Examples and Counterexamples

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✓ 12 results · all verified · 9 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The metrics d₁, d₂ and d_∞ on ℝⁿ are metrics and are Lipschitz equivalent, with explicit constants

Example

The discrete metric induces the discrete topology, in which every subset is clopen

Example

The supremum metric d_∞(f,g) = supₓ |f(x) - g(x)| on the bounded real-valued functions on a set

Example

The 2-adic absolute value gives an ultrametric on ℚ, in which every triangle is isosceles and every point of a ball is a centre

Example

The post-office metric d(x,y) = ‖x‖ + ‖y‖ for x ≠ y on ℝⁿ, and its isolated points

Example

min(|x-y|, 1) on ℝ has the usual topology and diameter at most 1

Counterexample

In {0} ∪ [1,2] with the metric of ℝ, the closure of B(0,1) = {0} is {0} while the closed ball is {0,1}

Counterexample

In the discrete metric the boundary of B(p,1) is empty while the sphere of radius 1 is everything but p

Counterexample

ℤ and {n + 1/n : n ≥ 2} are disjoint closed subsets of ℝ at distance 0, so the set-to-set distance is not a metric

Counterexample

On (0,∞) the metrics |x-y| and |1/x - 1/y| have the same topology and are not uniformly equivalent

Counterexample

On ℝ the metrics |x-y| and min(|x-y|,1) are uniformly but not Lipschitz equivalent

Counterexample

ℝ carries both an unbounded and a bounded metric inducing the same topology

Sources