definition 6.3 Closed set

open in the book · parts/02-mathematical-methods/04-topology.tex:39 · p. 193

ground object -- no derivation owed

Rests on

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 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 ¶

Supports

depends_on definition 12.70 Graph; closed and closable operators ¶
- depends_on proposition 12.73 The adjoint is always closed ¶
  - depends_on definition 12.78 Essential self-adjointness ¶
    - depends_on theorem A.266 von Neumann ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
    - depends_on theorem 12.80 von Neumann's criterion ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
      - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
  - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
    - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
depends_on definition 10.79 Parabolic boundary ¶
- 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 ¶
depends_on theorem 6.12 Heine–Borel in $\R^{N}$ ¶

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`	→	Union, intersection, difference	declared	parts/02-mathematical-methods/04-topology.tex:42
`depends_on`	→	Open set	declared	parts/02-mathematical-methods/04-topology.tex:42
`depends_on`	←	Graph; closed and closable operators	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1967
`depends_on`	←	Parabolic boundary	declared	parts/02-mathematical-methods/08-pdes.tex:2155
`depends_on`	←	Heine–Borel in $\R^{N}$	declared	parts/02-mathematical-methods/04-topology.tex:166