definition 6.9 Compact set

open in the book · parts/02-mathematical-methods/04-topology.tex:109 · p. 194

ground object -- no derivation owed

Rests on

depends_on definition 6.5 Open cover ¶
- depends_on definition 6.2 Open set ¶
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- depends_on equation 3.51 eq:set-indexed ¶ ↺

Supports

depends_on definition 7.142 Box-counting dimension ¶
- 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 proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
- depends_on remark 7.144 What is developed here, and what is not ¶
depends_on definition 7.127 Simple regions ¶
- depends_on lemma A.512 The boundary strip is thin ¶
  - depends_on theorem A.513 Change of variables ¶
    - depends_on corollary A.514 Degeneracy on a negligible set ¶
    - depends_on remark A.516 The hypotheses of the global form ¶
- depends_on remark A.516 The hypotheses of the global form ¶ ↺
- depends_on theorem 7.133 Gauss ¶
  - depends_on lemma A.708 The disturbance flux vanishes ¶
    - depends_on corollary A.711 The disturbance is a dipole at leading order ¶
  - depends_on lemma A.712 The force is a far-field integral ¶
  - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
    - depends_on theorem A.120 Stäckel and Robertson conditions ¶
      - depends_on example A.122 Cartesian ¶
      - depends_on example A.123 Circular cylindrical ¶
      - depends_on example A.124 Spherical ¶
      - depends_on proposition A.128 The ellipsoidal system separates ¶
  - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
  - depends_on lemma 31.11 Transport theorem for a material volume ¶
    - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on corollary 31.13 Incompressible flow ¶
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - depends_on theorem 31.22 Euler's equations of motion ¶
      - depends_on lemma A.712 The force is a far-field integral ¶ ↺
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶ ↺
      - depends_on proposition A.797 The outer limit loses a boundary condition ¶
      - depends_on proposition 31.80 Rayleigh's circulation criterion ¶
      - depends_on proposition 52.24 The p-mode cavity ¶ ↺
      - depends_on remark 31.23 Boundary conditions, and what the ideal model omits ¶
      - depends_on theorem 31.24 Bernoulli ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶
      - depends_on theorem 31.37 The Navier–Stokes equations ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶ ↺
      - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶
  - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
    - depends_on proposition 106.88 The obstruction ¶
      - depends_on theorem 106.95 The cancellation condition ¶
  - depends_on lemma 10.44 Green's identities ¶
    - depends_on proposition 10.71 Green's representation formula ¶
    - depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
      - depends_on example 10.49 The method of images ¶
      - depends_on lemma A.94 Fundamental solution of $\pp_{\bar z}$ ¶
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
      - depends_on proposition 10.71 Green's representation formula ¶ ↺
      - depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
    - depends_on theorem 10.45 Symmetry of the Green's function ¶
  - depends_on lemma 10.58 Darboux's equation for spherical means ¶
    - depends_on theorem 10.59 Kirchhoff's formula ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on theorem 10.69 Mean-value property ¶
      - depends_on corollary A.145 Harmonic functions are smooth ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
  - depends_on proposition 23.54 The geometrical amplitude diverges ¶
    - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on proposition 23.56 The phase jump at a fold ¶
      - depends_on remark 23.58 The caustics that are not turning surfaces ¶
    - depends_on remark 23.59 What is observed ¶
  - depends_on proposition 10.57 Energy in a backward cone ¶
    - depends_on proposition 28.63 Energy density, flux, and the conservation law ¶
  - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
    - depends_on definition 44.6 Stress–energy tensor ¶
      - depends_on definition 44.36 Energy conditions ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 28.42 The wave equation of a stretched string ¶
  - depends_on theorem 32.6 Evolution of phase volume ¶
    - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
    - depends_on definition 32.62 The Lorenz system ¶ ↺
    - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on phenomenon 32.68 Strange attractors ¶
    - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
  - … 1 more
- depends_on theorem 7.131 Green ¶
  - depends_on proposition 32.33 Bendixson's negative criterion ¶
  - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
  - depends_on theorem 7.132 Stokes ¶
    - depends_on definition 31.63 Circulation ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶ ↺
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
  - depends_on theorem 8.12 Cauchy ¶
    - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 106.1 The short-time kernel ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on corollary 17.76 Heaviside's expansion theorem ¶
      - depends_on proposition 17.77 Poles and stability ¶
    - depends_on lemma 100.27 Wick rotation ¶
      - depends_on lemma 100.28 The $d$-dimensional loop integral ¶
    - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
    - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
      - depends_on remark 28.26 Causality is visible in the susceptibility ¶
      - depends_on remark 28.62 Neither velocity is a speed limit ¶
    - depends_on theorem 17.94 Mellin inversion ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
  - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
    - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
  - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
    - depends_on example 10.97 Burgers shock ¶
    - depends_on example 10.98 Weak solutions are not unique ¶
    - depends_on proposition 10.102 Jump form of the entropy inequality ¶
depends_on definition A.498 Zero content ¶
- depends_on lemma A.499 What zero content buys ¶
  - depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
  - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
    - depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
    - depends_on lemma A.512 The boundary strip is thin ¶ ↺
  - depends_on theorem A.513 Change of variables ¶ ↺
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 32.8 Attractor and basin ¶
- depends_on definition 32.29 Limit cycle ¶
  - depends_on phenomenon 32.35 Self-sustained oscillation ¶
    - depends_on phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
      - depends_on definition 32.91 The circle map ¶
- depends_on definition 32.67 Strange attractor ¶
  - depends_on example 32.70 The Hénon map ¶
  - depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
  - depends_on phenomenon 32.68 Strange attractors ¶ ↺
- depends_on theorem 32.74 Oseledets, quoted ¶
  - depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
    - depends_on definition 32.67 Strange attractor ¶ ↺
depends_on definition 12.41 The operator classes ¶
- depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
  - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
    - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
  - depends_on theorem A.579 Stone–von Neumann ¶
  - depends_on theorem 12.91 Schur's lemma, commutant form ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
    - depends_on theorem 12.114 Stone–von Neumann ¶
  - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
- depends_on definition 12.58 Projection-valued measure ¶
  - depends_on definition 12.60 Functional calculus ¶
    - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on lemma A.247 Integration against a projection-valued measure ¶
  - depends_on theorem A.238 Spectral theorem, both forms ¶
    - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
    - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on example A.283 Momentum on the line ¶
    - depends_on theorem A.253 Stone ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on definition 12.60 Functional calculus ¶ ↺
    - depends_on theorem 12.107 Nuclear spectral theorem ¶
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
    - depends_on theorem 12.66 Stone ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
- 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 theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
  - depends_on definition 12.109 Weyl system ¶ ↺
  - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
    - depends_on proposition A.256 Density ¶
      - depends_on proposition A.259 The generator is self-adjoint ¶
  - depends_on lemma A.581 Composition law ¶ ↺
  - depends_on lemma A.582 Joint strong continuity ¶ ↺
  - 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 theorem A.253 Stone ¶ ↺
  - depends_on theorem 12.66 Stone ¶ ↺
  - depends_on theorem 25.34 Stone–von Neumann ¶ ↺
- depends_on lemma A.231 Restriction to an invariant closed subspace ¶
  - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
- depends_on lemma A.230 Sequential characterisation ¶
  - depends_on lemma A.232 Attainment ¶
    - depends_on lemma A.233 Construction of the system ¶ ↺
  - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
  - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶ ↺
- depends_on proposition 12.42 Elementary consequences ¶
  - depends_on lemma A.232 Attainment ¶ ↺
  - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    - depends_on lemma A.232 Attainment ¶ ↺
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on proposition A.241 The polynomial calculus is isometric ¶
    - depends_on theorem A.229 Hilbert–Schmidt ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
    - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
- depends_on theorem A.229 Hilbert–Schmidt ¶ ↺
- depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶ ↺
- depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
  - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
  - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
depends_on definition 13.66 Smooth action; free; proper; orbit ¶
- depends_on example 13.68 Why each hypothesis is there ¶
  - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
- depends_on lemma A.531 The projection is open ¶
  - 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 proposition A.532 Separation and countability of the quotient ¶ ↺
- depends_on lemma A.533 The orbit map has constant rank $d$ ¶
  - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
  - depends_on proposition A.535 Existence of a slice ¶
    - depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
    - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
- depends_on proposition A.535 Existence of a slice ¶ ↺
- depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
- depends_on theorem 13.67 Quotient manifold theorem ¶
  - depends_on example 13.68 Why each hypothesis is there ¶ ↺
  - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
    - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
  - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
    - depends_on example A.572 Rotational reduction of the central-force problem ¶
depends_on lemma A.506 Locality ¶
- depends_on proposition A.511 The substitution property is universal ¶
  - depends_on remark A.516 The hypotheses of the global form ¶ ↺
  - depends_on theorem A.513 Change of variables ¶ ↺
depends_on lemma A.505 A continuous partition of unity ¶
- depends_on lemma A.506 Locality ¶ ↺
depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
depends_on lemma A.307 Partition of unity on a compact manifold ¶
- depends_on definition A.308 Integral over the manifold ¶
  - depends_on theorem A.311 General Stokes theorem ¶
    - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
    - depends_on lemma A.555 The actions are well defined ¶
    - depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶
depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
… 11 more

Neighborhood

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partly declared
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not graded
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Open cover	declared	parts/02-mathematical-methods/04-topology.tex:112
`depends_on`	←	Box-counting dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4871
`depends_on`	←	Simple regions	declared	parts/02-mathematical-methods/05-real-analysis.tex:4235
`depends_on`	←	Zero content	declared	appendices/A-long-proofs.tex:24462
`depends_on`	←	Hausdorff; second countable; locally compact	declared	appendices/A-long-proofs.tex:25880
`depends_on`	←	Attractor and basin	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:219
`depends_on`	←	The operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1073
`depends_on`	←	Smooth action; free; proper; orbit	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3062
`depends_on`	←	Locality	declared	appendices/A-long-proofs.tex:24756
`depends_on`	←	A continuous partition of unity	declared	appendices/A-long-proofs.tex:24717
`depends_on`	←	Two elementary facts about compactness	declared	appendices/A-long-proofs.tex:25904
`depends_on`	←	Partition of unity on a compact manifold	declared	appendices/A-long-proofs.tex:15148
`depends_on`	←	Discrete subgroups of $\R^{f}$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6179
`depends_on`	←	Continuous images of compact sets	declared	parts/02-mathematical-methods/04-topology.tex:119
`depends_on`	←	Quoted: discrete subgroups of a real vector space	declared	appendices/A-long-proofs.tex:26689
`depends_on`	←	Riesz–Markov; quoted	declared	appendices/A-long-proofs.tex:12175
`depends_on`	←	Stone–Weierstrass; quoted	declared	appendices/A-long-proofs.tex:12100
`depends_on`	←	The direct method	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2184
`depends_on`	←	The horseshoe is a full shift, quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1097
`depends_on`	←	Poincaré–Bendixson, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:696
`depends_on`	←	Poincaré–Bendixson; quoted	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1268
`depends_on`	←	Heine–Borel on $\R$	declared	parts/02-mathematical-methods/04-topology.tex:136
`depends_on`	←	Heine–Borel in $\R^{N}$	declared	parts/02-mathematical-methods/04-topology.tex:166
`depends_on`	←	Compactness and sequential compactness	declared	parts/02-mathematical-methods/04-topology.tex:616