This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it's such a useful tool for talking about other things. Well, today, I'm going to show you a great example of that.
Last friday, I went through a fairly traditional approach to describing the topological product. The traditional approach not *very* difficult, but it's not particularly easy to follow either. The construction isn't really that difficult, but it's not easy to work out just what it all really means.
There is another approach to presenting it using category theory, and to me at least, it makes it a *whole* lot easier to grasp. To make the diagrams easier to draw, I'll adopt one shorthand: instead of writing (T,τ) for topological spaces, I'll use a single symbol, like **X**, with the understanding that **X** represents the *pair* of the set and the topology that form the topological space.
Suppose we have a set topological spaces, **E**1, **E**2, ..., **E**n. The product **P** = Πi=1..n**E**i is the *only* topological space with projection functions pi : **P** → **E**i, such that
for any other topological spaces **S**, if **S** has continuous functions fi : **S** → **E**i to each of the elements of the product, then there is *exactly one* continuous function g : **S** → **P** such that the following diagram commutes:
That's really just a repetition of the definition of categorical product, just made specific to the category **Top**. Everything I said in fridays post about what forms the open sets of the topological product space is directly implied by this categorical definition. The property of the open sets of the product topology being the coarsest structure of sets that maintains the structural properties of the product element topologies - that's implied by the categorical description.
To me, this is the real beauty of category theory, and the whole reason why I spent all that time explaining it. Being able to describe structures in the language of category theory makes things much easier to understand.