Mathematician Kasner (1878-1955) said he found it easier to teach topology to children than to adults because children "haven't been brain-washed by geometry".
LANCZOS ON TOPOLOGY In The Variational Principles of Mechanics, Cornelius Lanczos, 1948, pp. 13-14: "The position of a point in space is characterized by polar coordinates, (r, q, f). If these values are taken as rectangular coordinates of a point, we get a space whose geometrical properties are obviously greatly distorted compared with the actual space. Straight lines becomes curves, angles and distances are changed. Yet certain important properties of space remained unaltered by this mapping process. A point remains a point, the neighborhood of a point remains the neighborhood of a point, a curve remains a curve, adjacent curves remain adjacent curves. Continuous and differentiable curves remain continuous and differentiable curves. Now for the properties of the calculus of variations such 'topological' properties of space are the really important things, while the 'metrical' properties, such as distances, angles, areas, etc., are irrelevant."
A topology with a metric is a geometry.Topology exists without geometry, but geometry can't exist without topology.
Most of our daily decisions are topological, rather than geometric, but we are not taught this!
My Website describes topology inElsewhere, I note that we've had, not one Industrial Revolution< but many:
- mechanics
- thermodynamics
- electromagnetism
- general relativity
Thus, my files at this Website show that topology made possible our Industrial Revolutions!
- Mechanical Industrial Revolution, beginning in 12-13th centuries, with spread of thousands of watermills and windmills across Europe and Britain.
- Thermodynamic Industrial Revolution ("The Industrial Revolution"), beginning around 1776, with James Watt's development of his steam engine.
- Electrical Industrial Revolution, beginning aroudn 1880 with Edison wiring New York City for eletrical street lights.
- Electronic Industrial Revolution, early in 20th century with starts of radio stations.