theorem 6.11 Heine–Borel on $\R$

open in the book · parts/02-mathematical-methods/04-topology.tex:133 · p. 194

Rests on

depends_on axiom 7.1 Completeness of $\R$ ¶
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 ¶ ↺
proves proof ch:04-topology@proof-2 ¶

Supports

depends_on lemma A.365 Lebesgue number ¶
- depends_on lemma A.367 Homotopy lifting ¶
- depends_on lemma A.366 Unique path lifting ¶
  - depends_on lemma A.367 Homotopy lifting ¶ ↺
depends_on lemma A.136 Differentiation under the integral sign ¶
- depends_on proposition A.142 The mollification is smooth ¶
depends_on proposition 6.23 The punctured plane is not simply connected ¶
depends_on theorem 10.75 Weak maximum principle ¶
- depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
- depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
  - depends_on remark A.527 The hypotheses, and what happens without them ¶
- depends_on theorem 10.80 Parabolic maximum principle ¶
  - depends_on corollary 10.81 Uniqueness for the heat equation ¶
  - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶

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`	→	Completeness of $\R$	declared	parts/02-mathematical-methods/04-topology.tex:136
`depends_on`	→	Compact set	declared	parts/02-mathematical-methods/04-topology.tex:136
`depends_on`	←	Lebesgue number	declared	appendices/A-long-proofs.tex:17463
`depends_on`	←	Differentiation under the integral sign	declared	appendices/A-long-proofs.tex:7587
`depends_on`	←	The punctured plane is not simply connected	declared	parts/02-mathematical-methods/04-topology.tex:423
`depends_on`	←	Weak maximum principle	declared	parts/02-mathematical-methods/08-pdes.tex:2042
`proves`	←	ch:04-topology@proof-2	declared	parts/02-mathematical-methods/04-topology.tex:139