Topology · Chapter 02

Topological spaces

The 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.

(X, τ)

\emptyset, X \in τ

α ⋃ U_{α} \in τ

U \cap V \in τ

A

A^{\circ}

\partial A

X ∖ F

Lessons

Build the foundation

01From metrics to topologyChapter one proved three things about the open sets of a metric space. Those three facts now become the definition, and the distance that produced them is thrown away.Read lesson →
02Topological spacesWith the axioms in hand, a topology can be written down directly: name a set, name a collection of its subsets, and check three conditions. Spaces built this way need not come from any distance at all, and the small ones can be listed in full — which makes them the cheapest place to test a claim.Read lesson →
03Coarse and fineTwo topologies sit at the ends of the range. One declares every subset open, the other admits only the two the axioms force. Every other topology lies between them.Read lesson →
04Open sets of the line and planeThe line and the plane carry one topology so familiar that it is worth pinning down: the sets built from open intervals and open discs. Everything later in the course is tested against it, and it has a description that will become the idea of a basis — every open set is a union of the simple ones.Read lesson →
05Closed setsA set is closed when what is left of the space outside it is open. Nothing new is being assumed: every fact about closed sets is a fact about open ones, read through the complement.Read lesson →

06

Neighbourhoods and interiorA neighbourhood of a point is any set roomy enough around it to hold a whole open set containing it. Being open then turns out to be exactly the property of being a neighbourhood of every one of your own points. In a metric space the open set can always be taken to be a ball, which is how chapter 1 would have said it.

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07ClosureA set can be enlarged to a closed one, and there is a smallest way to do it. Add exactly the points that every open set around them already touches the original.Read lesson →

08Boundary and accumulation pointsThree kinds of point sit around a set. Inside, with room to spare; outside, with room to spare; and on the boundary, where every open set around the point meets both the set and its complement. A fourth question is asked separately: does the point have the set crowding it, even without counting the point itself? Those are the accumulation points, and they are what the closure adds.Read lesson →

09Chapter practiceEight questions across the whole chapter, with no teaching blocks. Deciding which idea a question is about is the part being practised here.Read lesson →

Mapping the lesson.

Topological spaces

Build the foundation