definition 6.15 Path-connected space

open in the book · parts/02-mathematical-methods/04-topology.tex:255 · 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 ¶ ↺

Supports

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 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 phenomenon 106.5 The Aharonov–Bohm effect ¶
      - depends_on theorem 106.16 Feynman–Kac ¶
  - 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 theorem 8.21 Laurent expansion ¶
      - depends_on definition 8.22 Isolated singularities; residue ¶
      - 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 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 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 ¶
depends_on proposition 6.16 prop:top-path-implies-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:259
`depends_on`	←	Path homotopy and the fundamental group	declared	appendices/A-long-proofs.tex:17408
`depends_on`	←	Simply connected space	declared	parts/02-mathematical-methods/04-topology.tex:283
`depends_on`	←	prop:top-path-implies-connected	declared	parts/02-mathematical-methods/04-topology.tex:264