ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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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.
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 map $(x,z) \mapsto x \cdot z$ on $\mathbb{R} \times \mathbb{R}$ and its transpose $z \mapsto (x \mapsto x \cdot z)$ traced through the exponential law

Example

Take $X := \mathbb{R}$ , $Z := \mathbb{R}$ and $Y := \mathbb{R}$ , all with 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 let

$f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}, \qquad f(x,z) := x \cdot z .$

Write $F := \Phi(f)$ for its transpose, so $F(z)(x) = x \cdot z$ : each $F(z)$ is the multiplication-by- $z$ map of $\mathbb{R}$ . This example checks every clause of the exponential law (The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$ , transposition is a bijection between $C(X \times Z, Y)$ and $C(Z, C(X,Y))$ with the compact-open topology) by hand on this pair:

$f$ is continuous on $\mathbb{R} \times \mathbb{R}$ with the product topology (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space);
each $F(z)$ is continuous, being Lipschitz with constant $|z|$ , so $F(z) \in C(\mathbb{R},\mathbb{R})$ ;
$F : \mathbb{R} \to C(\mathbb{R},\mathbb{R})$ is continuous for the compact-open topology, directly: if $|z - z_0| < \varepsilon/\iota(m)$ then $F(z) \in B_{[-m,m]}(F(z_0), \varepsilon)$ ;
$\mathbb{R}$ is locally compact, so the exponential law applies and $\Phi$ is a bijection $C(\mathbb{R} \times \mathbb{R}, \mathbb{R}) \to C(\mathbb{R}, C(\mathbb{R},\mathbb{R}))$ whose inverse returns $f$ from $F$ .

Claim 3 is the content of If $f : X \times Z \to Y$ is continuous then its transpose $F : Z \to C(X,Y)$ , $F(z)(x) = f(x,z)$ , is continuous for the compact-open topology, with no hypothesis on $X$ beyond being metric in this instance, verified without the tube lemma; claim 4 is where If $X$ is a locally compact metric space then the evaluation map is continuous for the compact-open topology is spent.

Facts & Assumptions

Given: $\mathbb{R}$ with $d(s,t) = |s-t|$ ; the product $\mathbb{R} \times \mathbb{R}$ with the product topology; the map $f(x,z) = xz$ and its transpose $F$ ; and for a natural $m \ge 1$ the interval $[-m,m] = \{\, t : -\iota(m) \le t \le \iota(m) \,\}$ (Intervals of $\mathbb{R}$ : the nine order-convex forms, nondegeneracy, and length, The canonical natural $\iota(n) = n \cdot 1_F$ of a field).

[L1]

The product topology on $\mathbb{R} \times \mathbb{R} = \prod_{k<2}\mathbb{R}$ is the metric topology of $d_\infty\big((x,z),(x',z')\big) = \max\{|x-x'|, |z-z'|\}$ (For $n \ge 1$ the product topology on $n$ copies of the usual topology of $\mathbb{R}$ is the metric topology of $d_\infty$ on $\mathbb{R}^n$ , and hence also of $d_1$ and $d_2$ , so $\mathbb{R}^n$ as a product and $\mathbb{R}^n$ as a metric space are one space, $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it, The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, 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, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, Maximum and minimum of a set, Every nonempty finite set of reals has a maximum and a minimum).

[L2]

$|uv| = |u||v|$ , $|u+v| \le |u|+|v|$ , and $|u| \ge 0$ (Basic properties of the absolute value, The triangle inequality, Absolute value in an ordered field).

[L3]

$\varepsilon$ - $\delta$ continuity of a map of metric spaces, and its agreement with continuity of the corresponding map of topological spaces (Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$ - $\delta$ form, Continuity of a map of topological spaces at a point and globally, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, Open ball, closed ball and sphere in a metric space, Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[L4]

A map Lipschitz with some constant is continuous (Lipschitz map, $\alpha$ -Hölder map for rational $0 < \alpha \le 1$ , and contraction, Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent, claims 2 and 3).

[L5]

$[-m,m]$ is a compact subset of $\mathbb{R}$ for every natural $m \ge 1$ , and every compact subset of $\mathbb{R}$ lies in some $[-m,m]$ ; the sets $B_K(g,\varepsilon)$ centred at $g$ are a neighbourhood base at $g$ in the compact-open topology on $C(\mathbb{R},\mathbb{R})$ (Heine-Borel in $\mathbb{R}^n$ : with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Open cover, subcover, compact metric space, and compact subset of a metric space, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, For a metric domain and a metric target the compact-open topology on $C(X,Y)$ is the topology of compact convergence, The topology of compact convergence on $C(X,Y)$ for metric $X$ and $Y$ : uniform convergence on each compact subset of $X$ , fact (U4), The compact-open topology on $C(X,Y)$ for a metric domain $X$ , with subbasis $S(K,V) = \{f : f[K] \subseteq V\}$ ).

[L6]

For every real $x$ there is a natural $m \ge 1$ with $|x| + 1 < \iota(m)$ , and $\iota(m) \ge 1 > 0$ for $m \ge 1$ (Every complete ordered field is Archimedean, Canonical naturals are positive and strictly increasing, For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ , The canonical natural $\iota(n) = n \cdot 1_F$ of a field).

[L7]

$\mathbb{R}$ is locally compact if every point has a compact set containing a ball around it; and then the exponential law holds for $X = \mathbb{R}$ and arbitrary $Z, Y$ , with $\Phi$ a bijection whose inverse sends $F$ to $(x,z) \mapsto F(z)(x)$ (Locally compact metric space: every point has a compact neighbourhood, The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$ , transposition is a bijection between $C(X \times Z, Y)$ and $C(Z, C(X,Y))$ with the compact-open topology, The evaluation map $e : C(X,Y) \times X \to Y$ , $e(f,x) = f(x)$ , Injection, surjection, bijection).

Verification

technique · direct

1.1

For claim 1, fix $(x_0,z_0) \in \mathbb{R} \times \mathbb{R}$ and a real $\varepsilon_0 > 0$ , and put $\delta := \min\{1,\ \varepsilon_0/(|x_0| + |z_0| + 1)\}$ , a real with $0 < \delta \le 1$ .

L1L2choose

1.2

For fixed $z$ the map $F(z) : x \mapsto xz$ satisfies $|xz - x'z| = |z||x-x'|$ , so it is Lipschitz with constant $|z|$ and continuous; this is claim 2.

L2L4

1.3

For claim 3, fix $z_0 \in \mathbb{R}$ and a neighbourhood $N$ of $F(z_0)$ in the compact-open topology; there are a compact $K$ and a real $\varepsilon > 0$ with $B_K(F(z_0),\varepsilon) \subseteq N$ , and a natural $m \ge 1$ with $K \subseteq [-m,m]$ , so $B_{[-m,m]}(F(z_0),\varepsilon) \subseteq B_K(F(z_0),\varepsilon) \subseteq N$ .

L5choose

1.4

For claim 4: given $x \in \mathbb{R}$ take a natural $m \ge 1$ with $|x| + 1 < \iota(m)$ ; then $[-m,m]$ is compact and $B(x,1) \subseteq [-m,m]$ , since $|t - x| < 1$ gives $|t| \le |x| + 1 < \iota(m)$ ; so $\mathbb{R}$ is a locally compact metric space.

L2L5L6

2.1

If $d_\infty\big((x,z),(x_0,z_0)\big) < \delta$ then $|x - x_0| < \delta$ and $|z - z_0| < \delta$ , so $|z| \le |z_0| + |z - z_0| < |z_0| + \delta$ .

step 1.1L1L2

2.2

Put $\eta := \varepsilon/\iota(m) > 0$ ; if $|z - z_0| < \eta$ then for every $x \in [-m,m]$ we get $|F(z)(x) - F(z_0)(x)| = |x||z - z_0| \le \iota(m)|z-z_0| < \iota(m)\eta = \varepsilon$ , so $F(z) \in B_{[-m,m]}(F(z_0),\varepsilon) \subseteq N$ .

step 1.3L2L6

3.1

Hence $|xz - x_0z_0| = |z(x - x_0) + x_0(z - z_0)| \le |z||x-x_0| + |x_0||z-z_0| < (|z_0| + \delta)\delta + |x_0|\delta = \delta\big(|x_0| + |z_0| + \delta\big) \le \delta\big(|x_0| + |z_0| + 1\big) \le \varepsilon_0$ .

step 1.1step 2.1L2

4.1

So $f$ is continuous at every point in the $\varepsilon$ - $\delta$ sense for the metric $d_\infty$ , hence continuous as a map of topological spaces for the product topology; this is claim 1.

step 3.1L1L3

5.1

As $N$ was an arbitrary neighbourhood of $F(z_0)$ and $z_0$ an arbitrary point, $F$ is continuous for the compact-open topology; this is claim 3, and it agrees with what If $f : X \times Z \to Y$ is continuous then its transpose $F : Z \to C(X,Y)$ , $F(z)(x) = f(x,z)$ , is continuous for the compact-open topology, with no hypothesis on $X$ beyond being metric gives from claim 1.

step 4.1step 1.3step 2.2L3L5

6.1

The exponential law therefore applies with $X = Z = Y = \mathbb{R}$ : transposition is a bijection between $C(\mathbb{R}\times\mathbb{R},\mathbb{R})$ and $C(\mathbb{R}, C(\mathbb{R},\mathbb{R}))$ , it sends the $f$ of claim 1 to the $F$ of claim 3, and its inverse sends $F$ back to $(x,z) \mapsto F(z)(x) = xz$ , which is $f$ ; this is claim 4.

step 4.1step 5.1step 1.4L7∎

Remarks

Claim 3 is proved twice on purpose. The general theorem If $f : X \times Z \to Y$ is continuous then its transpose $F : Z \to C(X,Y)$ , $F(z)(x) = f(x,z)$ , is continuous for the compact-open topology, with no hypothesis on $X$ beyond being metric derives it from the tube lemma, and the computation above derives it from a single estimate $|x||z-z_0| \le \iota(m)|z-z_0|$ on $[-m,m]$ . The second route is available here only because the compact sets of $\mathbb{R}$ are contained in intervals on which the first variable is bounded; the tube lemma is what replaces that boundedness in general.
What is not checked here, because it is not claimed. Nothing above asserts that the bijection $\Phi$ is a homeomorphism for any topology on the two sides. The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$ , transposition is a bijection between $C(X \times Z, Y)$ and $C(Z, C(X,Y))$ with the compact-open topology is a bijection of sets of continuous maps, and its own remark records what the homeomorphism form would additionally require.
The transposed family is a line of linear maps. As $z$ runs over $\mathbb{R}$ the functions $F(z)$ sweep out the multiplications by $z$ , and claim 3 says that this sweep is continuous when nearness of two such maps is measured uniformly on each bounded interval. It is not continuous for the uniform metric on all of $\mathbb{R}$ : for $z \ne z_0$ the difference $|x||z-z_0|$ exceeds $1$ once $|x|$ is large, so the uniform distance between $F(z)$ and $F(z_0)$ is $1$ whenever $z \ne z_0$ , which is the same phenomenon as the counterexample earlier on this page.

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Sources

Compact-open topology (Wikipedia) (standard reference, not scraped)
Exponential object (Wikipedia) (standard reference, not scraped)

The map (x,z)↦x⋅z(x,z) \mapsto x \cdot z(x,z)↦x⋅z on R×R\mathbb{R} \times \mathbb{R}R×R and its transpose z↦(x↦x⋅z)z \mapsto (x \mapsto x \cdot z)z↦(x↦x⋅z) traced through the exponential law