proposition 6.16 prop:top-path-implies-connected

open in the book · parts/02-mathematical-methods/04-topology.tex:262 · p. 196

Rests on

depends_on definition 6.13 Connected space ¶
- depends_on definition 3.36 Disjoint sets ¶
- 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 6.15 Path-connected space ¶
- 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 theorem 6.14 Intervals are connected ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on definition 6.13 Connected space ¶ ↺
- proves proof ch:04-topology@proof-4 ¶
proves proof ch:04-topology@proof-5 ¶

Nothing declares a dependency on this node yet.

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	Connected space	declared	parts/02-mathematical-methods/04-topology.tex:264
`depends_on`	→	Path-connected space	declared	parts/02-mathematical-methods/04-topology.tex:264
`depends_on`	→	Intervals are connected	declared	parts/02-mathematical-methods/04-topology.tex:264
`proves`	←	ch:04-topology@proof-5	declared	parts/02-mathematical-methods/04-topology.tex:267