False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31‡ rests on unproved material (inherited)

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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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‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Assuming choice, refuted: paracompactness is productive

Statement

Assuming the Axiom of Choice, paracompactness is productive.

Facts & Assumptions

Given: The Axiom of Choice and the lower-limit line $L$ .

[A1]

Choice implies countable choice: apply a choice function to any countably indexed family of nonempty sets (The Axiom of Choice, The Axiom of Countable Choice ( $\mathrm{AC}_\omega$ )).

[F1]

If $x<y$ in $L$ , then $[x,y)$ and $[y,y+1)$ are disjoint open neighbourhoods, so $L$ is Hausdorff (The lower-limit topology on $\mathbb{R}$ , with the half-open intervals $[a,b)$ as a basis, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).

[L1]

The lower-limit line is regular and Lindelöf; under countable choice every regular Lindelöf space is paracompact (The lower-limit line has a clopen basis, is regular, and is Lindelöf under countable choice, Under countable choice, every regular Lindelöf space is paracompact).

[L2]

Under choice, the product $L^2$ is not normal (Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square).

[L3]

The product of Hausdorff spaces is Hausdorff (Arbitrary products preserve $T_0$ , $T_1$ , and Hausdorffness).

[L4]

A paracompact Hausdorff space is normal (Every paracompact Hausdorff space is normal).

Refutation

technique · direct

1.1

By [A1] and [L1], both factors $L$ are paracompact.

A1L1

2.1

If paracompactness were productive, $L^2$ would be paracompact.

step 1.1

3.1

By [F1] and [L3], $L^2$ is Hausdorff; then [L4] would make it normal, contradicting [L2].

F1L2L3L4step 2.1

4.1

Hence the displayed productive assertion is refuted.

step 3.1∎

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: 142 results over 22 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

G. Gruenhage, General Topology Course Notes, Sorgenfrey plane and Jones's lemma (standard reference, not scraped)
R. Gardner, Notes on Munkres Section 41: Paracompactness (East Tennessee State University) (standard reference, not scraped)
Sorgenfrey topology (Encyclopedia of Mathematics) (standard reference, not scraped)