CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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 $K$ -topology on $\mathbb{R}$ is not uniformizable

Statement refuted

The $K$ -topology on $\mathbb R$ is uniformizable.

Facts & Assumptions

Given: The $K$ -topology on $\mathbb R$ .

[L1]

Every uniformizable space is regular (Every uniformizable space is regular).

[L2]

The $K$ -topology is Hausdorff and not regular (The $K$ -topology on $\mathbb{R}$ , generated by the open intervals together with their complements of $K = \{1/(n+1) : n \in \mathbb{N}\}$ , is $T_1$ and Hausdorff but not regular).

Counterexample

technique · contradiction

1.1

Suppose the $K$ -topology were uniformizable.

assume-contra

2.1

It would be regular by [L1].

step 1.1L1

3.1

This contradicts its nonregularity in [L2].

step 2.1L2

4.1

Therefore the supposition is false, and the $K$ -topology is not uniformizable.

step 3.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 17 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

R. Gardner, Introduction to Topology, notes on Munkres Section 31: The Separation Axioms (East Tennessee State University) (standard reference, not scraped)
M. Kunzinger, General Topology (standard reference, not scraped)

The KKK-topology on R\mathbb{R}R is not uniformizable