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

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Example

Let $X$ be any set and define $\delta : X \times X \to \mathbb{R}$ by

$\delta(x,y) := \begin{cases} 0 & \text{if } x = y, \\ 1 & \text{if } x \ne y. \end{cases}$

Then:

$\delta$ is a metric on $X$ (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric), the discrete metric.
For $x \in X$ and a real $r > 0$ : $B(x,r) = \begin{cases} \{x\} & 0 < r \le 1 \\ X & r > 1 \end{cases}, \qquad \bar B(x,r) = \begin{cases} \{x\} & 0 < r < 1 \\ X & r \ge 1 \end{cases}, \qquad S(x,1) = X \setminus \{x\}.$
Every subset of $X$ is open, hence every subset is closed, hence every subset is clopen (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). The metric topology of $\delta$ is the discrete topology, the collection of all subsets of $X$ .

The example is the standard source of counterexamples about balls: here the closed ball of radius $1$ is the whole space while the closure of the open ball of radius $1$ is a single point, and the sphere of radius $1$ is everything except the centre while the boundary of the ball is empty.

Facts & Assumptions

Given: A set $X$ , the function $\delta$ above, points $x, y, z \in X$ , a real $r > 0$ , and a subset $U \subseteq X$ .

[L1]

Metric axioms (M1), (M2), (M3) (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric); nonnegativity is a consequence and not needed as a hypothesis (Nonnegativity of a metric is a consequence of the other axioms, not an axiom).

[L2]

Balls: $B(x,r) = \{y : \delta(x,y) < r\}$ , $\bar B(x,r) = \{y : \delta(x,y) \le r\}$ and $S(x,r) = \{y : \delta(x,y) = r\}$ (Open ball, closed ball and sphere in a metric space).

[L3]

Open and closed: $U$ is open when every point of it has a ball around it inside it; $F$ is closed when $X \setminus F$ is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[L4]

Closed balls are closed (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed).

[L5]

Order: $0 < 1$ , so $0 \ne 1$ ; a sum of a positive and a nonnegative real is positive, and inequalities may be compared by trichotomy and transitivity (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Ordered field, Complete ordered field (least-upper-bound property)).

Verification

technique · direct

1.1

Separation and symmetry: $\delta(x,y) = 0$ holds exactly when $x = y$ , since the other value $1$ is different from $0$ ; and the defining clauses are unchanged when $x$ and $y$ are exchanged, since " $x = y$ " is.

L1L5

1.2

Triangle inequality: if $x = z$ then $\delta(x,z) = 0$ and the right side is a sum of two values in $\{0,1\}$ , hence at least $0$ ; and if $x \ne z$ then $y$ cannot equal both $x$ and $z$ , so at least one of $\delta(x,y), \delta(y,z)$ equals $1$ while the other is $0$ or $1$ , whence $\delta(x,y) + \delta(y,z) \ge 1 = \delta(x,z)$ .

L1L5

1.3

Balls: $\delta(x,x) = 0 < r$ always, so $x \in B(x,r)$ ; and for $y \ne x$ one has $\delta(x,y) = 1$ , so $y \in B(x,r)$ exactly when $1 < r$ , $y \in \bar B(x,r)$ exactly when $1 \le r$ , and $y \in S(x,1)$ always. This is claim 2.

L2L5

2.1

Every subset is open: for $U \subseteq X$ and $x \in U$ the ball $B(x,1)$ is $\{x\}$ by the computation of step 1.3, and $\{x\} \subseteq U$ .

L2L3L5

2.2

Claim 1 holds: $\delta$ satisfies (M1) and (M2) by step 1.1 and (M3) by step 1.2, so it is a metric on $X$ .

step 1.1step 1.2L1

3.1

Claim 3 holds: every subset is open by step 2.1, so for any $F \subseteq X$ the complement $X \setminus F$ is open and $F$ is closed; thus every subset is clopen and the metric topology is the full power set of $X$ . In particular each singleton $\{x\}$ is closed, in agreement with the closed-ball computation $\bar B(x,r) = \{x\}$ for $0 < r < 1$ and the fact that closed balls are closed.

step 1.3step 2.1L3L4

4.1

Claims 1, 2 and 3 hold by steps 2.2, 1.3 and 3.1.

step 1.3step 2.2step 3.1∎

Remarks

Every map out of a discrete space is continuous, since every preimage is open (For a map of metric spaces the following agree: $\varepsilon$ - $\delta$ continuity everywhere, preimages of open sets are open, preimages of closed sets are closed, sequential continuity, and $f(\overline{A}) \subseteq \overline{f(A)}$ ); so the discrete metric carries no information about $X$ beyond its cardinality, and is the extreme case at one end of the range of metrics on a set.
Convergence is eventual constancy. $x_k \to x$ in $(X,\delta)$ means $\delta(x_k,x) < 1$ eventually, that is $x_k = x$ for all large $k$ (Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$ ).
Boundedness is immediate: $X \subseteq B(x,2)$ for any $x$ , so every discrete metric space is bounded, with diameter $1$ as soon as $X$ has two points (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 10 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

Discrete space (Wikipedia) (standard reference, not scraped)
Metric space (Wikipedia) (standard reference, not scraped)
J. Munkres, Topology, 2nd ed., §12 (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)