definition 3.35 Union, intersection, difference

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:995 · p. 28

ground object -- no derivation owed

Rests on

depends_on definition 3.8 Conjunction ¶
depends_on definition 3.9 Disjunction ¶
depends_on definition 3.7 Negation ¶

Supports

depends_on definition 6.3 Closed set ¶
- depends_on definition 12.70 Graph; closed and closable operators ¶
  - depends_on proposition 12.73 The adjoint is always closed ¶
    - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on theorem A.266 von Neumann ¶
      - depends_on theorem 12.80 von Neumann's criterion ¶
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
- depends_on definition 10.79 Parabolic boundary ¶
  - depends_on theorem 10.80 Parabolic maximum principle ¶
    - depends_on corollary 10.81 Uniqueness for the heat equation ¶
    - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
- depends_on theorem 6.12 Heine–Borel in $\R^{N}$ ¶
depends_on definition 6.1 Topological space ¶
- depends_on definition A.529 Hausdorff; second countable; locally compact ¶
  - depends_on lemma A.530 Two elementary facts about compactness ¶
    - depends_on proposition A.532 Separation and countability of the quotient ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶
      - depends_on theorem A.537 The smooth structure on the orbit space ¶
- depends_on definition A.298 Half-space; smoothness on it ¶
  - depends_on definition A.299 Manifold with boundary ¶
    - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
      - depends_on definition A.309 The induced orientation of the boundary ¶
    - depends_on lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
- depends_on definition 11.33 Sigma-algebra ¶
  - depends_on definition 11.34 Measure; measure space ¶
    - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
      - depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶
    - depends_on definition 11.35 Null set; almost everywhere ¶
      - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶ ↺
      - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
    - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶ ↺
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition A.361 Covering map ¶
    - depends_on lemma A.366 Unique path lifting ¶
      - depends_on lemma A.367 Homotopy lifting ¶
    - depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
    - depends_on theorem A.364 Covering homomorphisms ¶
    - depends_on theorem A.363 Monodromy ¶
      - depends_on theorem A.364 Covering homomorphisms ¶ ↺
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
    - depends_on definition A.580 Weyl operator ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on lemma A.581 Composition law ¶
    - depends_on definition 12.109 Weyl system ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition A.580 Weyl operator ¶ ↺
      - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
      - depends_on theorem 12.114 Stone–von Neumann ¶
    - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on proposition A.256 Density ¶
    - depends_on lemma A.581 Composition law ¶ ↺
    - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - depends_on proposition A.256 Density ¶ ↺
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on theorem 12.66 Stone ¶
    - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
      - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem A.253 Stone ¶ ↺
    - depends_on theorem 12.66 Stone ¶ ↺
    - depends_on theorem 25.34 Stone–von Neumann ¶
      - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on theorem 25.38 Groenewold–van Hove ¶
  - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 13.50 Diffeomorphism ¶
    - depends_on definition 13.53 Immersion, submersion, embedding ¶
      - depends_on definition 13.54 Embedded submanifold ¶
      - depends_on example 13.55 An injective immersion that is not an embedding ¶
      - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
      - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
    - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
    - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
  - depends_on definition 6.15 Path-connected space ¶
    - depends_on definition A.362 Path homotopy and the fundamental group ¶
      - depends_on lemma A.368 The concatenation rules ¶
      - depends_on theorem A.363 Monodromy ¶ ↺
    - depends_on definition 6.17 Simply connected space ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on definition A.362 Path homotopy and the fundamental group ¶ ↺
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on example 6.18 ex:top-simply-connected ¶
      - depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
      - depends_on proposition 32.33 Bendixson's negative criterion ¶
      - depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
      - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
      - depends_on proposition 6.23 The punctured plane is not simply connected ¶
    - depends_on proposition 6.16 prop:top-path-implies-connected ¶
  - depends_on definition 6.17 Simply connected space ¶ ↺
  - depends_on lemma A.505 A continuous partition of unity ¶
    - depends_on lemma A.506 Locality ¶
      - depends_on proposition A.511 The substitution property is universal ¶
  - depends_on lemma A.531 The projection is open ¶
    - depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
    - depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
  - depends_on proposition 6.10 Continuous images of compact sets ¶
    - depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
  - depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
    - depends_on proposition 12.36 Boundedness is continuity ¶
      - depends_on theorem 12.74 Hellinger–Toeplitz ¶
- depends_on definition 6.26 Open ball; metric topology ¶
  - depends_on definition A.76 Star-shaped set ¶
    - depends_on lemma A.77 Poincaré lemma, converse form ¶
  - depends_on lemma 6.30 Lebesgue number ¶
    - depends_on theorem 6.31 Compactness and sequential compactness ¶
      - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on lemma A.488 Small chords cut off small arcs ¶
  - depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶ ↺
- depends_on definition 6.2 Open set ¶
  - depends_on definition A.361 Covering map ¶ ↺
  - depends_on definition A.141 Mollification ¶
    - depends_on proposition A.143 Reproduction identity ¶
  - depends_on definition A.134 Mean-value property ¶
    - depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
    - depends_on proposition A.143 Reproduction identity ¶ ↺
    - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on corollary A.146 Locally uniform limits of harmonic functions ¶ ↺
      - depends_on corollary A.145 Harmonic functions are smooth ¶
  - depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
  - depends_on definition 13.108 Star-shaped domain ¶
    - depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
      - depends_on example 13.110 Closed but not exact: the angle form ¶
      - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
  - depends_on definition 10.1 Partial differential equation; order ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
      - depends_on definition 10.93 Scalar conservation law ¶
      - depends_on definition 10.3 Principal part and principal symbol ¶
      - depends_on lemma 10.28 The separation constant ¶
      - depends_on proposition 28.44 Superposition ¶
      - depends_on proposition 10.15 Method of characteristics, first order ¶
      - depends_on proposition 10.43 Representation of the solution ¶
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
      - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on definition 10.24 Well-posed problem ¶
      - depends_on corollary A.152 Uniqueness ¶
      - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
      - depends_on corollary 10.81 Uniqueness for the heat equation ¶ ↺
      - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 10.25 The backward heat problem is ill posed ¶
      - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
  - depends_on definition 6.3 Closed set ¶ ↺
  - depends_on definition 6.13 Connected space ¶
    - depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
    - depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
    - depends_on theorem 6.14 Intervals are connected ¶
      - depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
  - depends_on definition 6.6 Continuous map ¶ ↺
  - depends_on definition 6.4 Neighbourhood ¶
  - depends_on definition 6.26 Open ball; metric topology ¶ ↺
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
      - depends_on definition 32.81 Correlation dimension ¶
      - depends_on example 32.80 The middle-thirds Cantor set ¶
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 7.142 Box-counting dimension ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on definition A.498 Zero content ¶
      - depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
      - depends_on definition 32.8 Attractor and basin ¶
      - depends_on definition 12.41 The operator classes ¶
      - depends_on definition 13.66 Smooth action; free; proper; orbit ¶
      - depends_on lemma A.506 Locality ¶ ↺
      - depends_on lemma A.505 A continuous partition of unity ¶ ↺
      - depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶ ↺
      - depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
      - … 11 more
    - depends_on lemma 6.30 Lebesgue number ¶ ↺
  - … 2 more
depends_on proposition 3.54 Preimages respect the set operations ¶
depends_on proposition 3.37 De Morgan's laws for sets ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Conjunction	declared	parts/02-mathematical-methods/01-logic-sets.tex:1008
`depends_on`	→	Disjunction	declared	parts/02-mathematical-methods/01-logic-sets.tex:1008
`depends_on`	→	Negation	declared	parts/02-mathematical-methods/01-logic-sets.tex:1008
`depends_on`	←	Closed set	declared	parts/02-mathematical-methods/04-topology.tex:42
`depends_on`	←	Topological space	declared	parts/02-mathematical-methods/04-topology.tex:30
`depends_on`	←	Preimages respect the set operations	declared	parts/02-mathematical-methods/01-logic-sets.tex:1371
`depends_on`	←	De Morgan's laws for sets	declared	parts/02-mathematical-methods/01-logic-sets.tex:1040