proposition A.532 Separation and countability of the quotient

open in the book · appendices/A-long-proofs.tex:25964 · p. 3052

Rests on

depends_on definition 13.66 Smooth action; free; proper; orbit ¶
- depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
- depends_on definition 13.49 Smooth map between manifolds ¶
  - depends_on definition 13.44 Atlas ¶ ↺
  - depends_on definition 13.45 Differentiable map on a topological space ¶
    - depends_on definition 13.44 Atlas ¶ ↺
    - depends_on definition 13.43 Coordinate system ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
depends_on lemma A.530 Two elementary facts about compactness ¶
- depends_on definition A.529 Hausdorff; second countable; locally compact ¶
  - depends_on definition 6.9 Compact set ¶ ↺
  - depends_on definition 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶ ↺
- depends_on definition 6.9 Compact set ¶ ↺
- depends_on proposition 6.10 Continuous images of compact sets ¶
  - depends_on definition 6.9 Compact set ¶ ↺
  - 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 ¶ ↺
  - proves proof ch:04-topology@proof-1 ¶
- proves proof app:A-long-proofs@proof-318 ¶
depends_on lemma A.531 The projection is open ¶
- depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
- depends_on definition 6.6 Continuous map ¶ ↺
- depends_on definition 6.2 Open set ¶ ↺
- proves proof app:A-long-proofs@proof-319 ¶
proves proof app:A-long-proofs@proof-320 ¶

Supports

depends_on remark A.542 Where each hypothesis is spent ¶
depends_on theorem A.537 The smooth structure on the orbit space ¶

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`	→	Smooth action; free; proper; orbit	declared	appendices/A-long-proofs.tex:25968
`depends_on`	→	Two elementary facts about compactness	declared	appendices/A-long-proofs.tex:25968
`depends_on`	→	The projection is open	declared	appendices/A-long-proofs.tex:25968
`depends_on`	←	Where each hypothesis is spent	declared	appendices/A-long-proofs.tex:26414
`depends_on`	←	The smooth structure on the orbit space	declared	appendices/A-long-proofs.tex:26224
`proves`	←	app:A-long-proofs@proof-320	declared	appendices/A-long-proofs.tex:25972