2026-09-04

Topological Scaffolding

As seen in Geometry of Data, we can construct a scaffolding to explain mathematics

1. Set
   1. These are the base of mathematics
2. Space
   1. Sets with some structure
3. Topology
   1. Space by virtue of defining open-sets
4. Metric
   1. A subset of Topology where a function called metric is defined and gives a metaphorical distance
   2. All metric spaces induce a valid topology
   3. All topologies don’t automatically have a valid metric
   4. We only care about topologies with a metric
5. Hausdorff Space
   1. A subset of topologies with disjoint neighborhoods
6. Manifolds
   1. Hausdorff spaces which are locally euclidean
7. Differentiable Manifold
   1. A manifold with an atlas that defines smoothness
8. Riemannian Manifold
   1. Differentiable manifold with a Reimannian metric defined

Related

20260904T164816-connection_bw_topology_and_abstract_algebra