SYSTEMATIC MATHEMATICS

Topology

This course contains 38 visible knowledge chapters in nine content-sized units. Which properties survive when a shape is stretched or bent without tearing or gluing? Begin with distance and familiar neighborhoods, compare metrics that do and do not preserve completeness, then remove the ruler and order topologies by their open information. Continuity is built from preimages and the pasting lemma; spaces are assembled by subspaces, products, quotients, labeled disjoint unions, and an explicit disk-boundary construction of the sphere. Connectedness is separated from local path behavior, while compactness gains its missing finite-product proof through the tube lemma. Separation becomes a continuous scale through Urysohn’s lemma, convergence beyond sequences is expressed by both nets and filters, and covering lifts compute the circle before the product theorem computes the torus fundamental group. Every symbol is defined before use, every theorem names its quantifiers, and every picture is converted into a proof or an explicit limitation.

Before this course: Completed Proof, Logic & Set Theory. The course assumes sets, functions, inverse images, equivalence relations, quantifiers, direct and contradiction proofs, induction, and counterexamples. Real Analysis is recommended for familiarity with metric limits and compact intervals, but every topological definition needed here is rebuilt locally. No measure theory, differential geometry, algebraic topology, category theory, manifolds, or computational topology is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Metrics turn distance into neighborhoods

Objective: Why can the theorem in “Metrics turn distance into neighborhoods” not be reused blindly in this boundary case: A nonnegative symmetric function without the triangle inequality need not support the ball-intersection proof. Name the first failed hypothesis or implication.

A metric d on X is nonnegative, symmetric, separates points, and satisfies d(x,z)≤d(x,y)+d(y,z); B_r(x)={y:d(x,y)<r}. This tells us which objects and open-set choices the chapter is about.

The result to establish is: The union of open balls defines a topology because arbitrary unions and finite intersections of ball-open sets remain open. The proof begins with “The empty set and X satisfy the ball-open condition immediately.” and closes with “For a finite intersection, take the smallest of the finitely many available radii.” Keep the carrier set, topology, arbitrary point or cover, and implication direction visible throughout.

The worked question is: Compare Euclidean, taxicab, and discrete metrics on a three-point set and list every open set. Begin with “On any finite metric space, each point has a positive minimum distance from all other points.” and finish with “Every subset is a union of singletons; Euclidean, taxicab, and discrete metrics therefore give the discrete topology here.” Then compare the result with this boundary: A nonnegative symmetric function without the triangle inequality need not support the ball-intersection proof.

The union of open balls defines a topology because arbitrary unions and finite intersections of ball-open sets remain open.

The empty set and X satisfy the ball-open condition immediately.

A point in a union inherits a ball from one member of that union.

For a finite intersection, take the smallest of the finitely many available radii.

Compare Euclidean, taxicab, and discrete metrics on a three-point set and list every open set.

  1. On any finite metric space, each point has a positive minimum distance from all other points.
  2. A ball of less than half that minimum contains only the chosen point, so every singleton is open.
  3. Every subset is a union of singletons; Euclidean, taxicab, and discrete metrics therefore give the discrete topology here.

Result: All eight subsets of {a,b,c} are open: ∅, the three singletons, the three two-point sets, and {a,b,c}.