proposition 6.23 The punctured plane is not simply connected

open in the book · parts/02-mathematical-methods/04-topology.tex:417 · p. 198

Rests on

depends_on definition 6.17 Simply 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 definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - 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 ¶ ↺
depends_on lemma 6.22 Nearby loops wind alike ¶
- depends_on definition 6.20 Winding number ¶
  - depends_on lemma 6.19 Continuous argument along a path ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - depends_on theorem 7.23 Intermediate value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on definition 7.20 Continuity at a point ¶
      - proves proof ch:05-real-analysis@proof-8 ¶
    - proves proof ch:04-topology@proof-6 ¶
- depends_on lemma 6.19 Continuous argument along a path ¶ ↺
- proves proof ch:04-topology@proof-7 ¶
depends_on theorem 6.11 Heine–Borel on $\R$ ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶ ↺
    - depends_on equation 3.51 eq:set-indexed ¶ ↺
- proves proof ch:04-topology@proof-2 ¶
proves proof ch:04-topology@proof-8 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Simply connected space	declared	parts/02-mathematical-methods/04-topology.tex:423
`depends_on`	→	Nearby loops wind alike	declared	parts/02-mathematical-methods/04-topology.tex:423
`depends_on`	→	Heine–Borel on $\R$	declared	parts/02-mathematical-methods/04-topology.tex:423
`proves`	←	ch:04-topology@proof-8	declared	parts/02-mathematical-methods/04-topology.tex:427