definition 6.17 Simply connected space

open in the book · parts/02-mathematical-methods/04-topology.tex:275 · p. 196

ground object -- no derivation owed

Rests on

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 ¶ ↺
depends_on definition 6.15 Path-connected space ¶
- depends_on definition 6.6 Continuous map ¶ ↺

Supports

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 phenomenon 106.5 The Aharonov–Bohm effect ¶
    - depends_on theorem 106.16 Feynman–Kac ¶
      - depends_on proposition 106.17 Euclidean time projects onto the ground state ¶
      - depends_on proposition 106.21 Reflection positivity is the positivity of the Hilbert space ¶
- depends_on theorem 8.16 Cauchy integral formula ¶
  - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on corollary A.723 Potential of one infinite row ¶
      - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - depends_on theorem 8.21 Laurent expansion ¶
    - depends_on definition 8.22 Isolated singularities; residue ¶
      - depends_on proposition 8.23 Residue at a simple pole ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on theorem 8.24 Residue theorem ¶ ↺
- 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 lemma A.368 The concatenation rules ¶
- depends_on theorem A.363 Monodromy ¶
  - depends_on theorem A.364 Covering homomorphisms ¶
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 23.55 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 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 ¶

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`	→	Continuous map	declared	parts/02-mathematical-methods/04-topology.tex:283
`depends_on`	→	Path-connected space	declared	parts/02-mathematical-methods/04-topology.tex:283
`depends_on`	←	Deformation of contours	declared	parts/02-mathematical-methods/06-complex-analysis.tex:388
`depends_on`	←	Path homotopy and the fundamental group	declared	appendices/A-long-proofs.tex:17408
`depends_on`	←	Field of extremals; slope function	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1866
`depends_on`	←	ex:top-simply-connected	declared	parts/02-mathematical-methods/04-topology.tex:295
`depends_on`	←	The $n$-sphere is simply connected for $n \ge 2$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1270
`depends_on`	←	Bendixson's negative criterion	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:747
`depends_on`	←	$\SO(3,\R)$ is not simply connected	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1199
`depends_on`	←	Bendixson–Dulac negative criterion	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1300
`depends_on`	←	The punctured plane is not simply connected	declared	parts/02-mathematical-methods/04-topology.tex:423