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Assuming choice, refuted: paracompactness is hereditary
Statement
Assuming the Axiom of Choice, paracompactness is hereditary.
Facts & Assumptions
Given: The Axiom of Choice and the ordinal spaces .
Choice implies the countable choice used by the ordinal compactness theorem (The Axiom of Choice).
Under countable choice, is countably compact and noncompact, while is compact (Every successor ordinal is compact in its order topology and every limit ordinal is not; and, assuming countable choice, is countably compact and sequentially compact while is compact).
Under choice, a countably compact paracompact Hausdorff space is compact (Assuming countable choice, every countably compact paracompact Hausdorff space is compact).
A compact space is paracompact (Every compact space is paracompact).
Every ordinal in its order topology is and Hausdorff, so each singleton is closed (Every ordinal with its order topology has a basis of clopen sets, and is , Hausdorff and regular, clauses 2 and 3).
Refutation
By [A1] and [L1], is compact, hence paracompact by [L3], and its initial segment is countably compact but noncompact.
The initial segment is open in , since its complement is the closed singleton consisting of the top endpoint.
If were paracompact, its Hausdorffness from [L4] would let [L2] make it compact, contradicting step 1.1.
Thus a paracompact space has the nonparacompact subspace , which refutes the displayed hereditary assertion.
Depends on
- Every compact space is paracompact
- Assuming countable choice, every countably compact paracompact Hausdorff space is compact
- Every successor ordinal is compact in its order topology and every limit ordinal is not; and, assuming countable choice, $\omega_1$ is countably compact and sequentially compact while $\omega_1 + 1$ is compact
- Every ordinal with its order topology has a basis of clopen sets, and is $T_1$, Hausdorff and regular
- The order topology on an ordinal, with the half-open intervals $(\alpha, \beta]$ and the initial segments $[0, \beta]$ as a basis
- Hereditary, open-hereditary and closed-hereditary properties of topological spaces
- The Axiom of Choice
Used by
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Sources
- First uncountable ordinal (Wikipedia) (standard reference, not scraped)
- G. Gruenhage, General Topology Course Notes (standard reference, not scraped)
- M. Aitken, Compactness notes (California State University San Marcos) (standard reference, not scraped)