Given a topological space , the diagonal embedding, or diagonal map of into (with the product topology) is the map

is homeomorphic to the image of (which is why we use the word “embedding”).

We can perform the same construction with objects other than topological spaces: for instance, there's a diagonal map , from a group into its direct sum with itself, given by the same formula. It's sensible to call this an embedding, too, since is a monomorphism.

We could also imagine a diagonal map into an n-fold product given by

Why call it the diagonal map?

Picture . Its diagonal embedding into the Cartesian plane is the diagonal line .

What's it good for?

Sometimes we can use information about the product space together with the diagonal embedding to get back information about . For instance, is Hausdorff if and only if the image of is closed in [proof]. If we know more about the product space than we do about , it might be easier to check if is closed than to verify the Hausdorff condition directly.

When studying algebraic topology, the fact that we have a diagonal embedding for any space lets us define a bit of extra structure in cohomology, called the cup product. This makes cohomology into a ring, so that we can bring additional algebraic muscle to bear on topological questions.

Another application from algebraic topology: there is something called an -space, which is essentially a topological space in which you can multiply two points together. The diagonal embedding, together with the multiplication, lets us say that the cohomology of an -space is a Hopf algebra; this structure lets us find out lots of things about -spaces by analogy to what we know about compact Lie groups.