definition 6.6 Continuous map

open in the book · parts/02-mathematical-methods/04-topology.tex:78 · p. 194

ground object -- no derivation owed

Rests on

depends_on definition 3.53 Preimage ¶
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 definition 6.1 Topological space ¶ ↺

Supports

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.590 The Gram matrix is universal ¶
      - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
  - 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 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.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
- 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 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 proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
    - 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 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 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 corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on theorem 13.133 Frobenius ¶
    - depends_on definition 13.58 Hypersurface ¶
      - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - depends_on theorem 13.59 Regular value theorem ¶
    - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
    - 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 theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺
  - 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.536 A slice is a chart domain downstairs ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶
  - depends_on lemma A.538 Submersions have smooth local sections ¶
    - depends_on proposition A.539 Universal property ¶
      - depends_on corollary A.540 Uniqueness of the smooth structure ¶
  - 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 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 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      - depends_on proposition 32.26 The Hopf normal form ¶
  - 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 106.3 Feynman's sum over histories ¶
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on theorem 8.21 Laurent expansion ¶
    - 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 example 8.25 A real integral by residues ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
      - depends_on theorem 31.68 Kutta–Joukowski ¶
      - depends_on theorem 9.26 Structure of the solutions ¶
  - 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 proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.53 A caustic point is a conjugate point ¶
    - depends_on remark 23.8 The action as a function, not a functional ¶
    - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶ ↺
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - 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 corollary A.514 Degeneracy on a negligible set ¶
      - depends_on remark A.516 The hypotheses of the global form ¶ ↺
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 remark A.542 Where each hypothesis is spent ¶ ↺
  - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
depends_on proposition 6.10 Continuous images of compact sets ¶
- depends_on lemma A.530 Two elementary facts about compactness ¶
  - depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
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 ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Preimage	declared	parts/02-mathematical-methods/04-topology.tex:86
`depends_on`	→	Open set	declared	parts/02-mathematical-methods/04-topology.tex:86
`depends_on`	→	Topological space	declared	parts/02-mathematical-methods/04-topology.tex:86
`depends_on`	←	Covering map	declared	appendices/A-long-proofs.tex:17384
`depends_on`	←	Strongly continuous one-parameter unitary group	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1749
`depends_on`	←	Homeomorphism	declared	parts/02-mathematical-methods/04-topology.tex:96
`depends_on`	←	Path-connected space	declared	parts/02-mathematical-methods/04-topology.tex:259
`depends_on`	←	Simply connected space	declared	parts/02-mathematical-methods/04-topology.tex:283
`depends_on`	←	A continuous partition of unity	declared	appendices/A-long-proofs.tex:24717
`depends_on`	←	The projection is open	declared	appendices/A-long-proofs.tex:25947
`depends_on`	←	Continuous images of compact sets	declared	parts/02-mathematical-methods/04-topology.tex:119
`depends_on`	←	$\varepsilon$–$\delta$ characterization	declared	parts/02-mathematical-methods/04-topology.tex:554