definition 6.7 Homeomorphism

open in the book · parts/02-mathematical-methods/04-topology.tex:89 · p. 194

ground object -- no derivation owed

Rests on

depends_on definition 3.47 Bijective map ¶
depends_on definition 3.51 Inverse map ¶
- depends_on definition 3.47 Bijective map ¶ ↺
- depends_on definition 3.49 Composition map ¶
- depends_on definition 3.50 Identity map ¶
depends_on definition 6.6 Continuous map ¶
- 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 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 corollary 13.134 Frobenius for a Pfaffian system ¶ ↺
      - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on theorem A.545 Liouville–Arnold ¶
  - depends_on definition 13.58 Hypersurface ¶
    - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - depends_on example A.294 The sphere, made explicit ¶
      - depends_on example 13.87 The $2$-sphere ¶
    - depends_on theorem 13.59 Regular value theorem ¶
      - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶ ↺
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶ ↺
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
  - 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 remark A.542 Where each hypothesis is spent ¶
    - 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 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 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 theorem A.537 The smooth structure on the orbit space ¶
    - 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 ¶ ↺

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`	→	Bijective map	declared	parts/02-mathematical-methods/04-topology.tex:96
`depends_on`	→	Inverse map	declared	parts/02-mathematical-methods/04-topology.tex:96
`depends_on`	→	Continuous map	declared	parts/02-mathematical-methods/04-topology.tex:96
`depends_on`	←	Diffeomorphism	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2571
`depends_on`	←	Immersion, submersion, embedding	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2634
`depends_on`	←	A slice is a chart domain downstairs	declared	appendices/A-long-proofs.tex:26180
`depends_on`	←	Hartman–Grobman, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:346
`depends_on`	←	Hartman–Grobman; quoted	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1223