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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 .
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 ()).
If in , then and are disjoint open neighbourhoods, so is Hausdorff (The lower-limit topology on , with the half-open intervals 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).
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).
Under choice, the product is not normal (Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square).
The product of Hausdorff spaces is Hausdorff (Arbitrary products preserve , , and Hausdorffness).
A paracompact Hausdorff space is normal (Every paracompact Hausdorff space is normal).
Refutation
By [A1] and [L1], both factors are paracompact.
If paracompactness were productive, would be paracompact.
By [F1] and [L3], is Hausdorff; then [L4] would make it normal, contradicting [L2].
Hence the displayed productive assertion is refuted.
Depends on
- 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
- Every paracompact Hausdorff space is normal
- Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square
- Arbitrary products preserve $T_0$, $T_1$, and Hausdorffness
- 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
- The Axiom of Choice
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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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)