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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Compact Hausdorff Baire is equivalent to DMC
Statement
Over , every compact Hausdorff space is a Baire space if and only if DMC holds (Dependent multiple choice in finite-level tree form, Baire space: a topological space in which every countable intersection of dense open subsets is dense, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).
Both directions are the content of the two preceding theorems of this page; this item records the equivalence and the exact form of each half, without adding any hypothesis of its own.
Facts & Assumptions
Given: The two implications proved earlier on this page.
proves that every compact Hausdorff space is Baire (DMC makes every compact Hausdorff space Baire).
Over , if every compact Hausdorff space is Baire then DMC holds (Compact Hausdorff Baire implies DMC).
The claim is the conjunction of the two implications of the statement, with no additional hypotheses (Baire space: a topological space in which every countable intersection of dense open subsets is dense, Dependent multiple choice in finite-level tree form).
Proof
Assume DMC; then by [F1] every compact Hausdorff space is Baire, which is the forward direction of the displayed equivalence.
Assume instead that every compact Hausdorff space is Baire; then by [F2] DMC holds, which is the reverse direction of the displayed equivalence.
The two implications hold unconditionally over , so the displayed biconditional is proved; the forward direction spends exactly DMC and the reverse direction spends only the Baireness hypothesis, as recorded by [F1] and [F2].
Depends on
- DMC makes every compact Hausdorff space Baire
- Compact Hausdorff Baire implies DMC
- Dependent multiple choice in finite-level tree form
- Baire space: a topological space in which every countable intersection of dense open subsets is dense
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
Used by
Dependency tree · two levels
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Sources
- David H. Fremlin, Dependent multiple choice and Baire's theorem (following Fossy and Morillon) (standard reference, not scraped)