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.
Assuming choice, paracompactness is not open-hereditary: inside
Statement refuted
Assuming the Axiom of Choice, every open subspace of a paracompact space is paracompact.
Facts & Assumptions
Given: The ordinal inclusion under the Axiom of Choice.
The space is not paracompact (Assuming choice, is countably compact, noncompact, and not paracompact).
Compact spaces are paracompact (Every compact space is paracompact).
Ordinal order topologies are , so their singleton subsets are closed (Every ordinal with its order topology has a basis of clopen sets, and is , Hausdorff and regular).
Counterexample
By [L2] and [L3], is paracompact.
Its subspace is open, as the complement consisting of the top endpoint is closed by [L4].
The open subspace is not paracompact by [L1], which refutes the displayed assertion.
Depends on
- Assuming choice, $\omega_1$ is countably compact, noncompact, and not paracompact
- Every compact space is paracompact
- 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
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: 103 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
- 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)