theorem 6.31 Compactness and sequential compactness

open in the book · parts/02-mathematical-methods/04-topology.tex:612 · p. 200

Rests on

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 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 equation 3.51 eq:set-indexed ¶ ↺
depends_on definition 6.29 Sequential compactness ¶
- depends_on definition 6.24 Metric ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶ ↺
depends_on lemma 6.30 Lebesgue number ¶
- depends_on definition 6.26 Open ball; metric topology ¶
  - depends_on definition 6.24 Metric ¶ ↺
  - depends_on definition 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶ ↺
- depends_on definition 6.5 Open cover ¶ ↺
- depends_on definition 6.29 Sequential compactness ¶ ↺
- proves proof ch:04-topology@proof-10 ¶
proves proof ch:04-topology@proof-11 ¶

Supports

depends_on lemma A.230 Sequential characterisation ¶
- depends_on lemma A.232 Attainment ¶
  - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
- depends_on lemma A.231 Restriction to an invariant closed subspace ¶
  - depends_on lemma A.233 Construction of the system ¶ ↺
- depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶ ↺
depends_on lemma A.488 Small chords cut off small arcs ¶
- depends_on theorem A.490 Equicontinuity of the normalised class ¶

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`	→	Compact set	declared	parts/02-mathematical-methods/04-topology.tex:616
`depends_on`	→	Sequential compactness	declared	parts/02-mathematical-methods/04-topology.tex:616
`depends_on`	→	Lebesgue number	declared	parts/02-mathematical-methods/04-topology.tex:616
`depends_on`	←	Sequential characterisation	declared	appendices/A-long-proofs.tex:11573
`depends_on`	←	Small chords cut off small arcs	declared	appendices/A-long-proofs.tex:23931
`proves`	←	ch:04-topology@proof-11	declared	parts/02-mathematical-methods/04-topology.tex:619