SWhat changed
Can a whole interval be assembled from countably many sets that leave a gap inside every smaller interval? Baire proves that it cannot. His 1899 thesis gives a nested-interval argument and introduces first and second category, a way to distinguish topologically small sets from a whole real interval. The framework also describes where suitable discontinuous functions remain continuous. It is a foundational principle about countable exceptional sets, rather than a statement about their total length.
Scientific scope
Can a whole interval be assembled from countably many sets that leave a gap inside every smaller interval? Baire proves that it cannot. His 1899 thesis gives a nested-interval argument and introduces first and second category, a way to distinguish topologically small sets from a whole real interval. The framework also describes where suitable discontinuous functions remain continuous. It is a foundational principle about countable exceptional sets, rather than a statement about their total length. The original theorem is stated for real intervals and, in the described extension, relative perfect subsets. It is not attributed here as an original theorem for every abstract complete metric or locally compact Hausdorff space. Historical non-dense means nowhere dense in this argument, not merely failure of density in the whole interval. Topological category is not Lebesgue measure, and a meagre set can be dense and have continuum cardinality. The common-continuity application is explicitly attributed in the original to Volterra in 1881; it is not claimed as Baire’s separate first discovery.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The original theorem is stated for real intervals and, in the described extension, relative perfect subsets. It is not attributed here as an original theorem for every abstract complete metric or locally compact Hausdorff space. Historical non-dense means nowhere dense in this argument, not merely failure of density in the whole interval. Topological category is not Lebesgue measure, and a meagre set can be dense and have continuum cardinality. The common-continuity application is explicitly attributed in the original to Volterra in 1881; it is not claimed as Baire’s separate first discovery.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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