False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Normality need not survive product with the interval

Statement

False: Every normal $T_{1}$ space $X$ has normal product $X \times [0, 1]$ , with the ordinary product topology and usual real interval.

Facts & Assumptions

Given: We refute the assertion under AC.

[F1]

For infinite $B \subseteq ω ∖ {0, 1}$ , the Rudin space $X_{R} (B)$ is $T_{1}$ , normal and not countably paracompact (Rudin ZFC Dowker space and its size).

[F2]

For $T_{1}$ spaces, normality of the interval product implies normality and countable paracompactness of the factor (Dowker product characterization).

[A1]

AC is assumed for both cited constructions (The Axiom of Choice).

Refutation

1.1

Set $B = {2, 3, 4, \dots}$ and take the specific witness $X = X_{R} (B)$ . Explicitly its points are functions $h (n) \leq ℵ_{n}$ whose coordinate cofinalities are all uncountable and strictly bounded by one finite aleph, with the relative ordinal box topology. The set $B$ is infinite and avoids zero and one, so F1 and A1 apply and verify that $X$ satisfies the asserted normality and $T_{1}$ hypotheses while failing countable paracompactness.

F1A1

2.1

If this $X \times [0, 1]$ were normal, F2 would imply that $X$ is countably paracompact, contradicting step 1.1. Thus the witness has a nonnormal interval product and refutes the universal assertion. The product in the conclusion is the ordinary product of the already defined space $X$ and the entire interval, including its endpoints. QED.

step 1.1F2A1

Depends on

The Axiom of Choice

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hart, Set-Theoretic Methods in General Topology, Chapter 4 Theorem 3.4 p. 28 and Chapter 6 pp. 35–38 (standard reference, not scraped)