Analysis -> (Topology) -> Algebra

Mathematics is divided into 2 major branches:
1. Analysis (Continuity, Calculus)
2. Algebra (Set, Discrete numbers, Structure)

In between the two branches, Poincaré invented in 1900s the Topology (拓扑学) – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.

Topology specialised in
–  ‘local knowledge’ = Point-Set Topology.
– ”global knowledge’ = Algebraic Topology.

Example:
The local data of consumer behavior uses ‘Point-Set Topology’; the global one is ‘BIG Data’ using Algebraic Topology.

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Hairy Ball Theorem

‘Hairy Ball Theorem’ (Topology)
1. “You can’t comb the hairs on a coconut.”
2. “There exists a calm point on earth where there is no wind.”
Both 1 & 2  are ‘Hairy Ball Theorem’.

If we have a ball with hairs sticking out from each point on it, then impossible to comb the hairs flat with a continuous movement without leaving at least one hair sticking up vertically (typically, at the center of a swirl). Not applicable to doughnut.
=> Wind pattern:
Ball = Earth
Combed down hairs = wind
Length of hairs = wind’s force
=> there must be a place on Earth where there is no wind at all.

Topology

Topology (by Poincaré)

Moniker “Rubber-Sheet Geometry“, compared with Geometry’s ‘rigid objects‘.

[Greek]= τοΠοζ(Place) λΟγια(Study)
[Latin]= Analysis Situs (Situation)

1. Remove (invariants) of geometry:

  • a. Euclidean (distance)
  • b. Affine (//, ratio)
  • c. Projective (cross-ratio)

2. Preserve ‘Neighbourhood’ (Nearness)

  • define ‘Continuity’ (Analysis)

3. Elastic deformation (stretch, bend, twist)

  • a line is no longer a line.