lemma 6.30 Lebesgue number
open in the book ·
parts/02-mathematical-methods/04-topology.tex:587
· p. 200
Rests on
-
depends_on
definition 6.26
Open ball; metric topology
¶
-
depends_on
definition 6.24
Metric
¶
- depends_on definition 3.43 Map ¶
- 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
¶
- depends_on definition 6.1 Topological space ¶ ↺
-
depends_on
definition 6.24
Metric
¶
-
depends_on
definition 6.5
Open cover
¶
- depends_on definition 6.2 Open set ¶ ↺
- depends_on equation 3.51 eq:set-indexed ¶ ↺
- depends_on definition 6.29 Sequential compactness ¶
- proves proof ch:04-topology@proof-10 ¶
Supports
-
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.234 The eigenvalues tend to zero, with finite multiplicity ¶
-
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.232
Attainment
¶
-
depends_on
lemma A.488
Small chords cut off small arcs
¶
- depends_on theorem A.490 Equicontinuity of the normalised class ¶
-
depends_on
lemma A.230
Sequential characterisation
¶
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 |
→ | Open ball; metric topology | declared | parts/02-mathematical-methods/04-topology.tex:592 |
depends_on |
→ | Open cover | declared | parts/02-mathematical-methods/04-topology.tex:592 |
depends_on |
→ | Sequential compactness | declared | parts/02-mathematical-methods/04-topology.tex:592 |
depends_on |
← | Compactness and sequential compactness | declared | parts/02-mathematical-methods/04-topology.tex:616 |
proves |
← | ch:04-topology@proof-10 | declared | parts/02-mathematical-methods/04-topology.tex:595 |