Given a topological space $X$ , the diagonal embedding, or diagonal map of $X$ into $X\times X$ (with the product topology) is the map $$x\stackrel{\Delta}{\longmapsto}(x,x) $$

$X$ is homeomorphic to the image of $\Delta$ (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 $\Delta\colon G\to G\times G$ , from a group into its direct sum with itself, given by the same formula. It's sensible to call this an embedding, too, since $\Delta$ is a monomorphism.

We could also imagine a diagonal map into an n-fold product given by $$x\stackrel{\Delta_n}{\longmapsto}(x,x,\ldots ,x)$$

Why call it the diagonal map?

Picture $\mathbb{R}$ . Its diagonal embedding into the Cartesian plane $\mathbb{R}\times\mathbb{R}$ is the diagonal line $y=x$ .

What's it good for?

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

When studying algebraic topology, the fact that we have a diagonal embedding for any space $X$ 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 $H$ -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 $H$ -space is a Hopf algebra; this structure lets us find out lots of things about $H$ -spaces by analogy to what we know about compact Lie groups.