Topology · Chapter 05

Product spaces

The 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.

X \times Y

π_{X} (x, y) = x

U \times V

π_{Y} (x, y) = y

i = 1 \prod n X_{i}

π_{i}

f = (f_{1}, f_{2})

R^{2}

Lessons

Build the foundation

01Cartesian productsTwo spaces can be combined into one whose points are pairs. The set is the Cartesian product, familiar from coordinates in the plane, and it comes with two maps that read off a coordinate. What it does not come with is a topology: that has to be chosen, and the choice is the subject of this chapter.Read lesson →
02The product topologyThe topology on a product is chosen to do one job: make the two projections continuous, and ask for nothing else. That means declaring the preimages of open sets under the projections to be open and taking the topology they generate — which is exactly what a subbasis is for.Read lesson →
03A basis for the productIntersecting two subbasic strips, one from each factor, gives an open box. The boxes are a basis for the product topology, which makes the product concrete: an open set of the plane is a union of open rectangles, exactly as it is a union of open discs.Read lesson →
04Maps into a productA map into a product is a pair of maps, one into each factor. The theorem of this chapter says the obvious thing is true: such a map is continuous exactly when both of its coordinates are. It is the reason the product topology is the one chosen, and it makes continuity of a map into the plane a question about two functions of one variable.Read lesson →
05Finite products of metric spacesA product of two metric spaces can be made metric again, and there are three natural ways to do it — add the distances, combine them under a root, or keep the larger. All three give the same open sets, and those open sets are exactly the product topology. So the plane as a metric space and the plane as a product are the same space.

Read lesson →

06Chapter practiceTen questions across the chapter, with no teaching blocks. Products are described three ways — by a subbasis, by a basis of boxes and by a metric — and choosing the convenient description is the skill being practised.Read lesson →

Mapping the lesson.

Product spaces

Build the foundation