lemma A.369 $\Phi$ is a two-sheeted covering

open in the book · appendices/A-long-proofs.tex:17752 · p. 2968

Rests on

depends_on definition A.361 Covering map ¶
- 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 6.2 Open set ¶ ↺
depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶
- depends_on definition 14.32 The special unitary group in two dimensions ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-11 ¶
depends_on theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$ ¶
- depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
  - depends_on equation 14.28 eq:lie-expso2 ¶
  - depends_on equation 14.25 eq:lie-so2-matrix ¶
  - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
    - depends_on definition 14.26 The rotation group ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
- depends_on proposition 14.36 Closed form of the exponential ¶
  - depends_on equation 14.28 eq:lie-expso2 ¶ ↺
  - depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶
    - depends_on definition 14.32 The special unitary group in two dimensions ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-13 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-14 ¶
- depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶ ↺
- depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-15 ¶
proves proof app:A-long-proofs@proof-224 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Covering map	declared	appendices/A-long-proofs.tex:17757
`depends_on`	→	$\SU(2)$ is the three-sphere	declared	appendices/A-long-proofs.tex:17757
`depends_on`	→	$\SU(2)$ is a two-to-one cover of $\SO(3,\R)$	declared	appendices/A-long-proofs.tex:17757
`proves`	←	app:A-long-proofs@proof-224	declared	appendices/A-long-proofs.tex:17760