lemma A.530 Two elementary facts about compactness

open in the book · appendices/A-long-proofs.tex:25894 · p. 3051

Rests on

depends_on definition A.529 Hausdorff; second countable; locally compact ¶
- 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 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 ¶

Supports

depends_on proposition A.532 Separation and countability of the quotient ¶
- 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`	→	Hausdorff; second countable; locally compact	declared	appendices/A-long-proofs.tex:25904
`depends_on`	→	Compact set	declared	appendices/A-long-proofs.tex:25904
`depends_on`	→	Continuous images of compact sets	declared	appendices/A-long-proofs.tex:25904
`depends_on`	←	Separation and countability of the quotient	declared	appendices/A-long-proofs.tex:25968
`proves`	←	app:A-long-proofs@proof-318	declared	appendices/A-long-proofs.tex:25908