definition 6.1 Topological space

open in the book · parts/02-mathematical-methods/04-topology.tex:20 · p. 193

ground object -- no derivation owed

Rests on

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 ¶

Supports

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 theorem A.311 General Stokes theorem ¶
  - 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 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 theorem 11.39 Birkhoff's pointwise ergodic theorem ¶ ↺
  - 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 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 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 A.592 The vacuum of the Schrödinger system ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
    - 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 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 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 proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
  - depends_on theorem A.253 Stone ¶
    - depends_on proposition A.263 The two constructions are inverse ¶
  - depends_on theorem 12.66 Stone ¶ ↺
  - depends_on theorem 25.34 Stone–von Neumann ¶ ↺
- 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 corollary 13.64 The image of a constant-rank map, locally ¶
      - depends_on definition 13.130 Distribution; involutive; integrable ¶
      - depends_on definition 13.58 Hypersurface ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
      - depends_on theorem 13.67 Quotient manifold theorem ¶
      - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
    - 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 proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
      - depends_on proposition A.535 Existence of a slice ¶
    - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.539 Universal property ¶
    - 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 proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - 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 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 definition A.362 Path homotopy and the fundamental group ¶ ↺
    - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on definition 23.52 Lagrangian family; caustic ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - 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 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 remark 29.12 The coordinate singularity, and what it is not ¶
    - 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 remark A.516 The hypotheses of the global form ¶
      - depends_on theorem A.513 Change of variables ¶
- 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 corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
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.232 Attainment ¶
      - 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 lemma A.488 Small chords cut off small arcs ¶
      - depends_on theorem A.490 Equicontinuity of the normalised class ¶
- 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.104 The recursion ¶
      - depends_on proposition A.110 Explicit solution of the majorant problem ¶
    - 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 A.150 Kruzhkov entropy pair ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶
    - depends_on definition 10.3 Principal part and principal symbol ¶
      - depends_on definition 10.10 Characteristic surface ¶
      - depends_on definition 10.4 The second-order operator ¶
      - depends_on definition 10.83 Weak solution ¶
    - depends_on lemma 10.28 The separation constant ¶
      - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
      - depends_on example 10.31 Cartesian separation ¶
      - depends_on example 10.32 Cylindrical separation ¶
      - depends_on example 10.33 Spherical separation ¶
      - depends_on proposition 23.9 Separation of the time ¶
      - depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
      - depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
    - depends_on proposition 28.44 Superposition ¶
    - depends_on proposition 10.15 Method of characteristics, first order ¶
      - depends_on definition 10.93 Scalar conservation law ¶ ↺
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
      - depends_on remark 31.90 Why a compression wave must steepen ¶
      - depends_on remark 23.3 What Part II owes this chapter ¶
    - 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 corollary 10.53 Duhamel for the wave equation ¶
  - 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 30.50 Love waves need a slow surface layer ¶
      - depends_on proposition 30.48 Rayleigh's secular equation ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
    - 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 theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; 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 12.78 Essential self-adjointness ¶
      - depends_on lemma A.260 Cayley transform of a 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 6.13 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 32.79 Box-counting dimension, restated from Part II ¶ ↺
      - 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 remark A.516 The hypotheses of the global form ¶ ↺
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 7.131 Green ¶
    - depends_on definition A.498 Zero content ¶
      - depends_on lemma A.499 What zero content buys ¶
    - depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
    - depends_on definition 32.8 Attractor and basin ¶
      - depends_on definition 32.29 Limit cycle ¶
      - depends_on definition 32.67 Strange attractor ¶
      - depends_on theorem 32.74 Oseledets, quoted ¶
    - depends_on definition 12.41 The operator classes ¶
      - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
      - depends_on definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
      - depends_on lemma A.230 Sequential characterisation ¶ ↺
      - depends_on proposition 12.42 Elementary consequences ¶
      - 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 definition 13.66 Smooth action; free; proper; orbit ¶
      - depends_on example 13.68 Why each hypothesis is there ¶
      - depends_on lemma A.531 The projection is open ¶ ↺
      - depends_on lemma A.533 The orbit map has constant rank $d$ ¶ ↺
      - 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 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

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Empty set	declared	parts/02-mathematical-methods/04-topology.tex:30
`depends_on`	→	Union, intersection, difference	declared	parts/02-mathematical-methods/04-topology.tex:30
`depends_on`	→	Subset	declared	parts/02-mathematical-methods/04-topology.tex:30
`depends_on`	→	eq:set-indexed	declared	parts/02-mathematical-methods/04-topology.tex:30
`depends_on`	←	Hausdorff; second countable; locally compact	declared	appendices/A-long-proofs.tex:25880
`depends_on`	←	Half-space; smoothness on it	declared	appendices/A-long-proofs.tex:14894
`depends_on`	←	Sigma-algebra	declared	parts/02-mathematical-methods/09-probability-statistics.tex:880
`depends_on`	←	Continuous map	declared	parts/02-mathematical-methods/04-topology.tex:86
`depends_on`	←	Open ball; metric topology	declared	parts/02-mathematical-methods/04-topology.tex:520
`depends_on`	←	Open set	declared	parts/02-mathematical-methods/04-topology.tex:36