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 . 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 (http://planetmath.org/ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed)]. 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.
|Date of creation||2013-03-22 14:20:41|
|Last modified on||2013-03-22 14:20:41|
|Last modified by||waj (4416)|