definition 6.2 Open set

open in the book · parts/02-mathematical-methods/04-topology.tex:33 · p. 193

ground object -- no derivation owed

Rests on

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 ¶

Supports

depends_on definition A.361 Covering map ¶
- depends_on lemma A.366 Unique path lifting ¶
  - depends_on lemma A.367 Homotopy lifting ¶
- depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
- depends_on theorem A.364 Covering homomorphisms ¶
- depends_on theorem A.363 Monodromy ¶
  - depends_on theorem A.364 Covering homomorphisms ¶ ↺
depends_on definition A.141 Mollification ¶
- depends_on proposition A.143 Reproduction identity ¶
depends_on definition A.134 Mean-value property ¶
- depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
- depends_on proposition A.143 Reproduction identity ¶ ↺
- depends_on theorem A.135 Converse of the mean-value property ¶
  - depends_on corollary A.146 Locally uniform limits of harmonic functions ¶ ↺
  - depends_on corollary A.145 Harmonic functions are smooth ¶
depends_on definition A.529 Hausdorff; second countable; locally compact ¶
- depends_on lemma A.530 Two elementary facts about compactness ¶
  - depends_on proposition A.532 Separation and countability of the quotient ¶
    - depends_on remark A.542 Where each hypothesis is spent ¶
    - depends_on theorem A.537 The smooth structure on the orbit space ¶
depends_on definition 13.108 Star-shaped domain ¶
- depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
  - depends_on example 13.110 Closed but not exact: the angle form ¶
  - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
depends_on definition 10.1 Partial differential equation; order ¶
- depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
  - depends_on definition A.103 First-order quasilinear system with zero data ¶
    - depends_on lemma A.104 The recursion ¶
      - depends_on lemma A.109 Domination ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on proposition A.110 Explicit solution of the majorant problem ¶
      - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
  - depends_on lemma A.105 Formal reduction of a normal problem ¶ ↺
- depends_on definition 10.2 Linear, semilinear, quasilinear ¶
  - depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
  - depends_on definition 10.93 Scalar conservation law ¶
    - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on lemma A.157 Two elementary identities ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on example 10.97 Burgers shock ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶
  - depends_on definition 10.3 Principal part and principal symbol ¶
    - depends_on definition 10.10 Characteristic surface ¶
      - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on definition 10.4 The second-order operator ¶
      - depends_on definition 10.10 Characteristic surface ¶ ↺
      - depends_on definition 10.42 Green's function ¶
      - depends_on definition 10.6 Type of a second-order operator ¶
    - depends_on definition 10.83 Weak solution ¶
      - depends_on definition 10.87 Weak form of an elliptic problem ¶
      - depends_on definition 10.85 Sobolev space ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
      - depends_on proposition 10.84 Consistency ¶
  - depends_on lemma 10.28 The separation constant ¶
    - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
      - depends_on theorem A.120 Stäckel and Robertson conditions ¶
    - depends_on example 10.31 Cartesian separation ¶
      - depends_on example A.122 Cartesian ¶
    - depends_on example 10.32 Cylindrical separation ¶
      - depends_on example A.123 Circular cylindrical ¶
    - depends_on example 10.33 Spherical separation ¶
      - depends_on example A.124 Spherical ¶
    - depends_on proposition 23.9 Separation of the time ¶
      - depends_on proposition 23.10 The characteristic function is the abbreviated action ¶
      - depends_on remark 23.12 Two functions, two roles ¶
    - depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
      - depends_on example 10.38 The vibrating string ¶
      - depends_on phenomenon 28.53 Chladni's figures ¶
      - depends_on theorem 10.36 Solution of the heat problem on an interval ¶
    - depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
      - depends_on remark 23.19 The same conditions govern the Helmholtz equation ¶
  - depends_on proposition 28.44 Superposition ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶
    - depends_on definition 10.93 Scalar conservation law ¶ ↺
    - depends_on proposition 10.16 Cauchy's characteristic strips ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
    - depends_on remark 31.90 Why a compression wave must steepen ¶
    - depends_on remark 23.3 What Part II owes this chapter ¶
  - depends_on proposition 10.43 Representation of the solution ¶
  - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
  - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on corollary 10.53 Duhamel for the wave equation ¶
- depends_on definition 10.24 Well-posed problem ¶
  - depends_on corollary A.152 Uniqueness ¶
  - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
  - depends_on corollary 10.81 Uniqueness for the heat equation ¶
  - depends_on definition 30.22 Boundary conditions of elastostatics ¶
    - depends_on proposition 30.50 Love waves need a slow surface layer ¶
    - depends_on proposition 30.48 Rayleigh's secular equation ¶
      - depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
    - depends_on proposition 30.23 Virtual work; the weak form ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on proposition 30.34 Minimum of the potential energy ¶
      - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
  - depends_on proposition 10.25 The backward heat problem is ill posed ¶
  - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
    - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
depends_on definition 6.3 Closed set ¶
- 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 theorem 12.80 von Neumann's criterion ¶
    - 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 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 $\R^{N}$ ¶
depends_on definition 6.13 Connected space ¶
- depends_on proposition 6.16 prop:top-path-implies-connected ¶
- depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
- depends_on theorem 6.14 Intervals are connected ¶
  - depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
depends_on definition 6.6 Continuous map ¶
- depends_on definition A.361 Covering map ¶ ↺
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
  - depends_on definition A.580 Weyl operator ¶
    - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
  - depends_on definition 12.109 Weyl system ¶
    - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
    - depends_on definition A.580 Weyl operator ¶ ↺
    - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
    - depends_on theorem A.579 Stone–von Neumann ¶ ↺
    - depends_on theorem 12.114 Stone–von Neumann ¶
  - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
    - depends_on proposition A.256 Density ¶
      - depends_on proposition A.259 The generator is self-adjoint ¶
  - depends_on lemma A.581 Composition law ¶ ↺
  - depends_on lemma A.582 Joint strong continuity ¶ ↺
  - depends_on proposition A.256 Density ¶ ↺
  - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
    - depends_on theorem 12.66 Stone ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
      - depends_on definition 12.109 Weyl system ¶ ↺
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
      - depends_on theorem 25.34 Stone–von Neumann ¶
  - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
  - depends_on theorem A.253 Stone ¶
    - depends_on proposition A.263 The two constructions are inverse ¶
  - depends_on theorem 12.66 Stone ¶ ↺
  - depends_on theorem 25.34 Stone–von Neumann ¶ ↺
- depends_on definition 6.7 Homeomorphism ¶
  - depends_on definition 13.50 Diffeomorphism ¶
  - depends_on definition 13.53 Immersion, submersion, embedding ¶
    - depends_on definition 13.54 Embedded submanifold ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
      - depends_on definition 13.130 Distribution; involutive; integrable ¶
      - depends_on definition 13.58 Hypersurface ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
      - depends_on theorem 13.67 Quotient manifold theorem ¶
      - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
    - depends_on example 13.55 An injective immersion that is not an embedding ¶
    - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
      - depends_on proposition A.535 Existence of a slice ¶
    - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.539 Universal property ¶
    - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
    - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶
    - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on definition 32.22 Bifurcation ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on example 32.18 A linear centre that is really a stable focus ¶
  - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
- depends_on definition 6.15 Path-connected space ¶
  - depends_on definition A.362 Path homotopy and the fundamental group ¶
    - depends_on lemma A.368 The concatenation rules ¶
    - depends_on theorem A.363 Monodromy ¶ ↺
  - depends_on definition 6.17 Simply connected space ¶
    - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 106.1 The short-time kernel ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on definition A.362 Path homotopy and the fundamental group ¶ ↺
    - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on definition 23.52 Lagrangian family; caustic ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - depends_on example 6.18 ex:top-simply-connected ¶
      - depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
    - depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶ ↺
    - depends_on proposition 32.33 Bendixson's negative criterion ¶
    - depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
      - depends_on remark 29.12 The coordinate singularity, and what it is not ¶
    - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
    - depends_on proposition 6.23 The punctured plane is not simply connected ¶
  - depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
- depends_on definition 6.17 Simply connected space ¶ ↺
- depends_on lemma A.505 A continuous partition of unity ¶
  - depends_on lemma A.506 Locality ¶
    - depends_on proposition A.511 The substitution property is universal ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem A.513 Change of variables ¶
- depends_on lemma A.531 The projection is open ¶
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
  - depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
- depends_on proposition 6.10 Continuous images of compact sets ¶
  - depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
- 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 ¶
depends_on definition 6.4 Neighbourhood ¶
depends_on definition 6.26 Open ball; metric topology ¶
- 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.231 Restriction to an invariant closed subspace ¶
      - 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 definition 6.5 Open cover ¶
- depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
  - depends_on definition 32.81 Correlation dimension ¶
  - depends_on example 32.80 The middle-thirds Cantor set ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 7.142 Box-counting dimension ¶
    - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶ ↺
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
    - depends_on remark 7.144 What is developed here, and what is not ¶
  - depends_on definition 7.127 Simple regions ¶
    - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on theorem A.513 Change of variables ¶ ↺
    - depends_on remark A.516 The hypotheses of the global form ¶ ↺
    - depends_on theorem 7.133 Gauss ¶
      - depends_on lemma A.708 The disturbance flux vanishes ¶
      - depends_on lemma A.712 The force is a far-field integral ¶
      - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
      - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
      - depends_on lemma 10.44 Green's identities ¶
      - depends_on lemma 10.58 Darboux's equation for spherical means ¶
      - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 10.57 Energy in a backward cone ¶
      - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on theorem 32.6 Evolution of phase volume ¶
      - … 1 more
    - depends_on theorem 7.131 Green ¶
      - depends_on proposition 32.33 Bendixson's negative criterion ¶ ↺
      - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶ ↺
      - depends_on theorem 7.132 Stokes ¶
      - depends_on theorem 8.12 Cauchy ¶
      - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶ ↺
  - depends_on definition A.498 Zero content ¶
    - depends_on lemma A.499 What zero content buys ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on theorem A.513 Change of variables ¶ ↺
  - depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
  - depends_on definition 32.8 Attractor and basin ¶
    - depends_on definition 32.29 Limit cycle ¶
      - depends_on phenomenon 32.35 Self-sustained oscillation ¶
    - depends_on definition 32.67 Strange attractor ¶
      - depends_on example 32.70 The Hénon map ¶
      - depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
      - depends_on phenomenon 32.68 Strange attractors ¶
    - depends_on theorem 32.74 Oseledets, quoted ¶
      - depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
  - depends_on definition 12.41 The operator classes ¶
    - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
      - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on theorem A.579 Stone–von Neumann ¶ ↺
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
    - depends_on definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on lemma A.247 Integration against a projection-valued measure ¶
      - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
    - depends_on lemma A.230 Sequential characterisation ¶ ↺
    - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on lemma A.232 Attainment ¶ ↺
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    - depends_on theorem A.229 Hilbert–Schmidt ¶
    - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
      - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
      - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
  - depends_on definition 13.66 Smooth action; free; proper; orbit ¶
    - depends_on example 13.68 Why each hypothesis is there ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
    - depends_on lemma A.531 The projection is open ¶ ↺
    - depends_on lemma A.533 The orbit map has constant rank $d$ ¶ ↺
    - depends_on proposition A.535 Existence of a slice ¶ ↺
    - depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
    - depends_on theorem 13.67 Quotient manifold theorem ¶ ↺
  - depends_on lemma A.506 Locality ¶ ↺
  - depends_on lemma A.505 A continuous partition of unity ¶ ↺
  - depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
  - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on definition A.308 Integral over the manifold ¶
      - depends_on theorem A.311 General Stokes theorem ¶
  - depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
  - … 11 more
- depends_on lemma 6.30 Lebesgue number ¶ ↺
… 2 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Topological space	declared	parts/02-mathematical-methods/04-topology.tex:36
`depends_on`	←	Covering map	declared	appendices/A-long-proofs.tex:17384
`depends_on`	←	Mollification	declared	appendices/A-long-proofs.tex:7779
`depends_on`	←	Mean-value property	declared	appendices/A-long-proofs.tex:7541
`depends_on`	←	Hausdorff; second countable; locally compact	declared	appendices/A-long-proofs.tex:25880
`depends_on`	←	Star-shaped domain	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5166
`depends_on`	←	Partial differential equation; order	declared	parts/02-mathematical-methods/08-pdes.tex:72
`depends_on`	←	Closed set	declared	parts/02-mathematical-methods/04-topology.tex:42
`depends_on`	←	Connected space	declared	parts/02-mathematical-methods/04-topology.tex:224
`depends_on`	←	Continuous map	declared	parts/02-mathematical-methods/04-topology.tex:86
`depends_on`	←	Neighbourhood	declared	parts/02-mathematical-methods/04-topology.tex:48
`depends_on`	←	Open ball; metric topology	declared	parts/02-mathematical-methods/04-topology.tex:520
`depends_on`	←	Open cover	declared	parts/02-mathematical-methods/04-topology.tex:59
`depends_on`	←	Lebesgue number	declared	appendices/A-long-proofs.tex:17463
`depends_on`	←	The projection is open	declared	appendices/A-long-proofs.tex:25947