Topology

Topology is the plumbing of calculus.

The key concept of topology is a neighbourhood.

It allows defining limit and continuity, without the need to specify a precise numerical value to the distance between two points with a metric.

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Table of contents | 72, 2k, 54
- 1. Covering space | 84, 84, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 1.1. Double cover | 🔗 link | 🗖 nosplit | ↑ parent "Covering space"
- 2. Neighbourhood (mathematics) | 12 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
- 3. Topological space | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
- 4. Manifold | 164, 424, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 4.1. Atlas (topology) | 25, 25, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
    - 4.1.1. Coordinate chart | 🔗 link | 🗖 nosplit | ↑ parent "Atlas"
  - 4.2. Covariant derivative | 39 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
  - 4.3. Tangent space | 136, 148, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
    - 4.3.1. Tangent vector to a manifold | 12 | 🔗 link | 🗖 nosplit | ↑ parent "Tangent space"
  - 4.4. One-form | 48 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
- 5. Metric (mathematics, $d (x, y)$ ) | 96, 319, 9 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 5.1. Metric space | 10, 223, 8 | 🔗 link | 🗖 nosplit | ↑ parent "Metric"
    - 5.1.1. Metric space vs normed vector space vs inner product space | 23 | 🔗 link | 🗖 nosplit | ↑ parent "Metric space"
    - 5.1.2. Normed vector space | 0, 49, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Metric space"
      - 5.1.2.1. Inner product space | 23, 49, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Normed vector space"
        
        5.1.2.1.1. Inner product | 26 | 🔗 link | 🗖 nosplit | ↑ parent "Inner product space"
    - 5.1.3. Norm (mathematics, |x|) | 31, 97, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Metric space"
      - 5.1.3.1. Norm induced by an inner product | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Norm"
      - 5.1.3.2. Metric induced from a norm | 41 | 🔗 link | 🗖 nosplit | ↑ parent "Norm"
    - 5.1.4. Complete metric space | 44 | 🔗 link | 🗖 nosplit | ↑ parent "Metric space"
- 6. Compact space | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
- 7. Dense set | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
- 8. Simply connected space | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
- 9. Homotopy | 0, 526, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 9.1. Generalized Poincaré conjecture | 282, 526, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Homotopy"
- 10. Real coordinate space ( $R^{n}$ ) | 0, 89, 21 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 10.1. Real line ( $R^{1}$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space"
  - 10.2. Real plane ( $R^{2}$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space"
  - 10.3. Real coordinate space of dimension three ( $R^{3}$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space"
  - 10.4. Real coordinate space of dimension four ( $R^{4}$ , 4D) | 10, 28, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space"
    - 10.4.1. Visualizing 4D | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space of dimension four"
  - 10.5. Dimension | 0, 4, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space"
    - 10.5.1. Infinite dimensional | 4, 4, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Dimension"
      - 10.5.1.1. Finite dimensional | 🔗 link | 🗖 nosplit | ↑ parent "Infinite dimensional"
  - 10.6. Euclidean space | 23, 57, 12 | 🔗 link | 🗖 nosplit | ↑ parent "Real coordinate space"
    - 10.6.1. Cartesian coordinate system | 🔗 link | 🗖 nosplit | ↑ parent "Euclidean space"
    - 10.6.2. Polar coordinate system | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Euclidean space"
      - 10.6.2.1. Spherical coordinate system | 🔗 link | 🗖 nosplit | ↑ parent "Polar coordinate system"
    - 10.6.3. Pythagorean theorem | 🔗 link | 🗖 nosplit | ↑ parent "Euclidean space"
    - 10.6.4. Non-Euclidean geometry | 0, 34, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Euclidean space"
      - 10.6.4.1. Elliptic geometry | 0, 34, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Non-Euclidean geometry"
        
        10.6.4.1.1. Model of elliptic geometry | 0, 34, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Elliptic geometry"
        
        10.6.4.1.1.1. Projective elliptic geometry | 34 | 🔗 link | 🗖 nosplit | ↑ parent "Model of elliptic geometry"
      - 10.6.4.2. Hyperbolic gemoetry | 0, 0, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Non-Euclidean geometry"
        
        10.6.4.2.1. Hyperbolic functions | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Hyperbolic gemoetry"
        
        10.6.4.2.1.1. Hyperbolic sine | 🔗 link | 🗖 nosplit | ↑ parent "Hyperbolic functions"
        
        10.6.4.2.1.2. Hyperbolic cossine | 🔗 link | 🗖 nosplit | ↑ parent "Hyperbolic functions"

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Ancestors

Calculus | 17, 8k, 203
Mathematics | 17, 28k, 633
Ciro Santilli's Homepage | 262, 218k, 4k

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Incoming links

Covering space | 84, 84, 1
Neighbourhood | 12