Topology · Chapter 04

Continuity and homeomorphisms

The one condition that survives without a distance, the ways of checking it, and what it means for two spaces to be the same shape.

f^{- 1} (V) \in τ_{X}

f : X \to Y

X ≅ Y

g \circ f

f ∣_{A}

V \in τ_{Y}

f^{- 1}

id_{X}

Lessons

Build the foundation

01Continuous mapsA map is continuous when pulling any open set back through it leaves an open set. No distance appears, no closeness is measured, and nothing is said about what the map does to an open set going forwards.Read lesson →
02Continuity at a pointContinuity was defined for a map as a whole. It can also be asked at a single point: whatever tolerance is demanded around the value, there is room around the point that the map sends inside it. A map is continuous exactly when it is continuous at every one of its points, and this local form is the one that recovers the epsilon and delta of the real line.Read lesson →
03Checking continuity on a basisTesting every open set of the target is impossible for the line, where there are as many open sets as there are subsets of the rationals. Two shortcuts make continuity checkable: test only a basis — or even a subbasis — and test closed sets instead if they are more convenient.Read lesson →
04Compositions and restrictionsContinuity survives the operations that build maps out of maps. Composing two continuous maps gives a continuous one — proved in the first lesson of this chapter — and so does restricting a continuous map to a subspace, or narrowing its target to a subspace holding the image. What needs care is gluing: two continuous maps that agree where their pieces meet make a continuous whole, provided the pieces are both closed or both open.Read lesson →
05HomeomorphismsTwo spaces are the same, topologically, when there is a bijection between their points that matches their open sets exactly. Such a map is a homeomorphism, and it is the right notion of sameness here: it is continuous in both directions, so nothing stated in open sets can tell the two spaces apart.

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06Topological invariantsProving two spaces homeomorphic means producing a map. Proving they are not means finding a property that one has and the other lacks, and that any homeomorphism would have to carry across. Such a property is called a topological invariant, and collecting them is how the rest of the course is used.Read lesson →

07Chapter practiceTwelve questions across the chapter, with no teaching blocks. Continuity has three equivalent tests and two directions, and deciding which one a question is really about is the practice here.Read lesson →

Mapping the lesson.

Continuity and homeomorphisms

Build the foundation