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Topology | Math Infinitum

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Topology

Mathematics · Course 03

Topology

From distances to spaces without them: open sets, continuity, compactness, connectedness and the shapes gluing builds.

(X, τ)

U \in τ

f^{- 1} (V)

B_{r} (x)

A

X \times Y

X / \sim

S^{1}

\partial A

α ⋃ U_{α}

Course map

9 chapters

01Metric spaces and convergenceWhat a distance has to do, the balls and open sets it produces, and what convergence means once the only tool is a distance.
02Topological spacesThe three rules that survive when the distance is dropped, the spaces they allow, and what interior, closure and boundary mean with nothing but open sets.
03Bases and subspacesDescribing a topology by a small family that generates it, and the topology a subset inherits from the space around it.
04Continuity and homeomorphismsThe one condition that survives without a distance, the ways of checking it, and what it means for two spaces to be the same shape.
05Product spacesThe topology a product of spaces carries, the basis of boxes that describes it, and why a map into a product is continuous exactly when its coordinates are.

CompactnessThe property that thins every open cover: what it means for a subset, what it does to a continuous map, and what it amounts to in a metric space and in the line.

↗

07ConnectednessBeing in one piece, said in open sets rather than in journeys: separations, connected subsets, what a continuous map does to them, paths, and the pieces a space falls into.↗

08Separation and countabilityHow much a topology can tell apart, and how little of it needs writing down: the separation ladder from T0 to normal, what Hausdorff buys, and the two countability axioms the familiar spaces satisfy.↗

09Quotients and constructionsBuilding spaces by gluing: the topology a quotient carries, the rule that recognises the result, and the circle, cylinder, torus, projective plane and one-point compactification it builds.↗