definition 3.31 Empty set

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• 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 12.70 Graph; closed and closable operators ¶
        • depends_on proposition 12.73 The adjoint is always closed ¶
      • depends_on definition 10.79 Parabolic boundary ¶
        • depends_on theorem 10.80 Parabolic maximum principle ¶
      • depends_on theorem 6.12 Heine–Borel in $\R^{N}$ ¶
    • 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