definition 6.26 Open ball; metric topology

open in the book · parts/02-mathematical-methods/04-topology.tex:509 · p. 199

ground object -- no derivation owed

Rests on

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 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶
- depends_on definition 3.38 Ordered pair and Cartesian product ¶ ↺
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 3.8 Conjunction ¶
    - depends_on definition 3.9 Disjunction ¶
    - depends_on definition 3.7 Negation ¶
  - depends_on definition 3.30 Subset ¶
  - depends_on equation 3.51 eq:set-indexed ¶
depends_on definition 6.1 Topological space ¶ ↺

Supports

depends_on definition A.76 Star-shaped set ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶
depends_on lemma 6.30 Lebesgue number ¶
- depends_on theorem 6.31 Compactness and sequential compactness ¶
  - 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.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 ¶
depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
- depends_on proposition 12.36 Boundedness is continuity ¶
  - depends_on theorem 12.74 Hellinger–Toeplitz ¶
    - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶

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`	→	Metric	declared	parts/02-mathematical-methods/04-topology.tex:520
`depends_on`	→	Open set	declared	parts/02-mathematical-methods/04-topology.tex:520
`depends_on`	→	Topological space	declared	parts/02-mathematical-methods/04-topology.tex:520
`depends_on`	←	Star-shaped set	declared	appendices/A-long-proofs.tex:4931
`depends_on`	←	Lebesgue number	declared	parts/02-mathematical-methods/04-topology.tex:592
`depends_on`	←	$\varepsilon$–$\delta$ characterization	declared	parts/02-mathematical-methods/04-topology.tex:554