Alphabeta MathHow 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.
Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word
Definition
A topological space is paracompact when every open cover of has an open refinement which covers and is locally finite. In symbols, for every open cover there is a locally finite open cover such that every lies in some .
No separation axiom is included in this definition. Some sources reserve the word paracompact for the conjunction of this covering property with Hausdorffness. Here the covering property is named by itself, and any use of Hausdorffness is stated explicitly.
Remarks
The finite-subcover condition defining compactness is recalled in Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right. A finite family is locally finite, but compactness and paracompactness remain distinct definitions because their conclusions quantify over different refinements of a cover.
Depends on
Used by
- False: BPI proves Stone's theorem for metric spaces False statement
- Assuming countable choice, every countably compact paracompact Hausdorff space is compact Lemma
- Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses Lemma
- Corson's rational metric space is not metacompact Lemma
- Every paracompact Hausdorff space is regular Lemma
- Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements {Vₛ} and {Wₛ} with Vₛ⊆ Wₛ subseteqWₛ⊆ Uₛ Lemma
- Under countable choice, every regular Lindelöf space is paracompact Lemma
- Every closed subspace of a paracompact space is paracompact Proposition
- Every compact space is paracompact Proposition
- Choice and convention ledger for paracompactness, Stone's theorem, and partitions of unity Remark
- Every paracompact Hausdorff space is normal Theorem
- For a Hausdorff space, paracompactness is equivalent, under choice and dependent choice, to the existence of a locally finite subordinate partition of unity for every open cover Theorem
- Relative consistency of BPI with failure of Stone's theorem Theorem
- Relative consistency of DC with failure of Stone's theorem Theorem
- Stone's theorem, under choice: every metric space is paracompact Theorem
- Under choice, a space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable Theorem
Dependency tree · two levels
8 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
- Dartmouth Point-Set Topology, Lecture 25 (standard reference, not scraped)
- R. Gardner, Notes on Munkres Section 41: Paracompactness (East Tennessee State University) (standard reference, not scraped)