A 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.

Three surveyors measure the same field with different instruments and disagree about every number, while agreeing exactly about which plots have room to spare.

A ball of the maximum metric is a square, and a ball of the Euclidean metric is the disc inscribed in it. Each fits inside the other around any of its points, so the two metrics have the same open sets — and those are the boxes' unions, the product topology.

On $X\times Y$ with metrics $d_X$ and $d_Y$, each of the rules below is a metric. All three give the same topology, and that topology is the product topology. In particular $\mathbb{R}\times\mathbb{R}$ as a product is the plane with its usual metric.

• $d_\infty\le d_2\le d_1\le2d_\infty$, so a ball of any one of them contains a ball of each of the others around the same point: the same open sets.
• A ball of $d_\infty$ is a box: $B\big((x,y),r\big)=B(x,r)\times B(y,r)$. That single identity is why the metric topology and the product topology agree.
• Convergence in the product is convergence in each coordinate, which follows at once from the maximum metric.
• A finite product of metric spaces is metrisable. A countable product is too, with a cleverer metric; an uncountable product of non-trivial spaces is not, which is one reason the course stops at finite products.
• The distance used on $\mathbb{R}^n$ throughout the course is $d_2$, and the chapter on metrics showed the three agree there as well.

A pair is within r of another in the maximum metric exactly when each coordinate is within r of its partner, which is membership of the box of the two balls. So every ball of the maximum metric is a basic open box, and every box contains such a ball around each of its points, by taking the smaller of the two radii. The two bases fit inside each other, so the topologies are equal.

The maximum of two numbers is below r exactly when both are.

So a ball of the maximum metric is exactly a box of two balls.

Conversely, inside a basic box each coordinate has a ball inside its side.

Taking the smaller radius gives a ball of the maximum metric inside the box.

Each basis fits inside the other around every point, so the two topologies are the same.

Add them, combine them under a root, or keep the larger. All three are metrics on the product.

• $\max$, sum and root all define metrics on a product, and all three give one topology.
• A ball of the maximum metric is a box of two balls.
• That topology is the product topology, so the plane as a product is the plane as a metric space.
• Convergence in a product is convergence in each coordinate.

Great work! You now know how to:

• combine two metrics into a metric on the product;
• prove that a maximum-metric ball is a box;
• use that the product and metric topologies agree;
• test convergence in a product coordinate by coordinate.