lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1265 · p. 556

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 example 6.18 ex:top-simply-connected ¶
- depends_on definition 6.17 Simply connected space ¶ ↺
depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
- depends_on proposition 7.22 Sequential characterization ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-7 ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶
  - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
    - depends_on definition A.36 Cut ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶ ↺
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-32 ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-4 ¶
- proves proof ch:05-real-analysis@proof-10 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-12 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Simply connected space	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1270
`depends_on`	→	ex:top-simply-connected	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1270
`depends_on`	→	Heine–Cantor: uniform continuity	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1270
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-12	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1273