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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Nowhere dense, meagre, residual, and comeagre subsets of a topological space
Definition
Let be a topological space and let . The set is nowhere dense when (Interior, closure, boundary, exterior, derived set and isolated point in a topological space). It is meagre when there is a sequence of nowhere dense subsets of with . It is residual, or comeagre, when is meagre. The empty union shows that is meagre, including when .
Depends on
Used by
- Under Dependent Choice, nowhere differentiable functions form a residual subset of C([0,1],ℝ) Corollary
- Add, cov, non and cof for null and meagre ideals Definition
- Borel master codes for null and meagre sets Definition
- Shelah's universal-meagre forcing Definition
- The property of Baire Definition
- A universal-meagre stage absorbs an old nowhere-dense tree Example
- FALSE: the rational numbers form a Baire space False statement
- A universal-meagre generic absorbs old nowhere-dense sets Lemma
- Baire diagonal passage from finite regularity to smooth metrics Lemma
- Borel-code, measure, category, and perfect-set absoluteness Lemma
- Category-game strategies characterize meagreness and local comeagreness Lemma
- Critical images of proper local Fredholm restrictions are nowhere dense Lemma
- Elementary bounds on ideal cardinal invariants Lemma
- Null and meagre master codes are cofinal Lemma
- Transfer of null and meagre invariants between Cantor space and the line Lemma
- Uniform closed nowhere-dense covers for Borel meagre sections Lemma
- Open subspaces and residual subspaces of Baire spaces are Baire Proposition
- The meagre subsets of a topological space form a sigma-ideal Proposition
- Critical images of proper local Fredholm restrictions are nowhere dense externally Remark
- Equivalent forms of the Baire property Theorem
- MA makes unions of fewer than continuum many meagre sets meagre Theorem
- Morse--Smale metrics are residual for a fixed Morse function Theorem
- Sard--Smale residual regular values for Fredholm maps Theorem
Dependency tree · two levels
9 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
- David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
- Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
- MFF General Topology course summary, §4.3 (standard reference, not scraped)
- Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)