ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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$\min(|x-y|, 1)$ on $\mathbb{R}$ has the usual topology and diameter at most $1$

Example

On $\mathbb{R}$ , let $d_{\mathbb{R}}(x,y) = |x-y|$ be the usual metric (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$ , and it is unbounded) and put

$\rho(x,y) := \min\{\, |x - y|,\ 1 \,\} \qquad (x, y \in \mathbb{R}).$

Then:

$\rho$ is a metric on $\mathbb{R}$ , uniformly equivalent and therefore topologically equivalent to $d_{\mathbb{R}}$ (Topologically, uniformly and Lipschitz equivalent metrics on a set, Lipschitz equivalence implies uniform equivalence implies topological equivalence); so $(\mathbb{R},\rho)$ has exactly the usual topology of the real line.
$(\mathbb{R},\rho)$ is a bounded metric space and $\operatorname{diam}(\mathbb{R}) = 1$ in the metric $\rho$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space), whereas $(\mathbb{R}, d_{\mathbb{R}})$ is not bounded at all and has no diameter.

This is the concrete instance of $\min(d,1)$ and $d/(1+d)$ are metrics uniformly equivalent to $d$ , so every metric space carries a bounded metric with the same topology on the real line, and it is the witness used for the failure of two plausible-sounding claims: that boundedness is topological, and that topologically equivalent metrics are Lipschitz equivalent.

Facts & Assumptions

Given: The real line with $d_{\mathbb{R}}(x,y) = |x-y|$ and the function $\rho = \min\{d_{\mathbb{R}}, 1\}$ ; the set $D := \{\, \rho(x,y) : x, y \in \mathbb{R} \,\}$ .

[L1]

$d_{\mathbb{R}}$ is a metric on $\mathbb{R}$ and $(\mathbb{R}, d_{\mathbb{R}})$ is not bounded, so it has no diameter (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$ , and it is unbounded, Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

[L2]

For any metric $d$ , the function $\min\{d,1\}$ is a metric, is bounded with diameter at most $1$ on a nonempty space, and is uniformly equivalent to $d$ ( $\min(d,1)$ and $d/(1+d)$ are metrics uniformly equivalent to $d$ , so every metric space carries a bounded metric with the same topology).

[L3]

Uniform equivalence implies topological equivalence (Lipschitz equivalence implies uniform equivalence implies topological equivalence, Topologically, uniformly and Lipschitz equivalent metrics on a set).

[L4]

The minimum of a two-element set of reals exists and is one of them (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set); $|1 - 0| = 1$ and $0 < 1$ (Basic properties of the absolute value, Absolute value in an ordered field, The multiplicative identity is positive).

[L5]

The diameter is the least upper bound of the set of distances, so it is $\ge$ every distance and $\le$ every upper bound of them; and it is unique (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, Suprema and infima are unique, Complete ordered field (least-upper-bound property), Ordered field).

Verification

technique · direct

1.1

By [L1] the function $d_{\mathbb{R}}$ is a metric on $\mathbb{R}$ and $\mathbb{R}$ is not bounded in it.

1.2

$\rho(0,1) = \min\{|0-1|,\ 1\} = \min\{1,1\} = 1$ , so $1 \in D$ .

2.1

By [L2] applied to $d = d_{\mathbb{R}}$ : $\rho$ is a metric on $\mathbb{R}$ , the space $(\mathbb{R},\rho)$ is bounded, $\operatorname{diam}(\mathbb{R}) \le 1$ in $\rho$ , and $\rho$ is uniformly equivalent to $d_{\mathbb{R}}$ .

step 1.1L2

3.1

Claim 2: the diameter of $\mathbb{R}$ in $\rho$ is an upper bound of $D$ and $1 \in D$ by step 1.2, so it is $\ge 1$ ; combined with $\le 1$ from step 2.1 this gives $\operatorname{diam}(\mathbb{R}) = 1$ in the metric $\rho$ , while by step 1.1 the space has no diameter at all in $d_{\mathbb{R}}$ .

step 1.1step 1.2step 2.1L5

3.2

Claim 1: $\rho$ is a metric uniformly equivalent to $d_{\mathbb{R}}$ by step 2.1, hence topologically equivalent to it, so the metric topology of $\rho$ is the usual topology of $\mathbb{R}$ .

step 2.1L3

4.1

Claims 1 and 2 hold by steps 3.2 and 3.1.

step 3.1step 3.2∎

Remarks

Nothing about $\mathbb{R}$ is special here except unboundedness. The same computation on any unbounded metric space gives a bounded metric with the same topology ( $\min(d,1)$ and $d/(1+d)$ are metrics uniformly equivalent to $d$ , so every metric space carries a bounded metric with the same topology); the real line is chosen because it is the space every reader already has.
The diameter is exactly $1$ and not less, which is what step 1.2 contributes: for the construction $\min\{d,1\}$ the bound of $\min(d,1)$ and $d/(1+d)$ are metrics uniformly equivalent to $d$ , so every metric space carries a bounded metric with the same topology is attained as soon as $d$ takes some value $\ge 1$ , because the new metric then takes the value $1$ itself.
The two metrics are not Lipschitz equivalent, so this pair also witnesses the strictness of the first implication of Lipschitz equivalence implies uniform equivalence implies topological equivalence; that computation is On $\mathbb{R}$ the metrics $|x-y|$ and $\min(|x-y|,1)$ are uniformly but not Lipschitz equivalent.

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 65 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Equivalence of metrics (Wikipedia) (standard reference, not scraped)
Metric space (Wikipedia) (standard reference, not scraped)
J. Munkres, Topology, 2nd ed., §20 (standard reference, not scraped)
R. Gardner, Introduction to Topology, notes on Munkres Section 20: The Metric Topology (East Tennessee State University) (standard reference, not scraped)

min⁡(∣x−y∣,1)\min(|x-y|, 1)min(∣x−y∣,1) on R\mathbb{R}R has the usual topology and diameter at most 111