Topologies as Categories Okay, so we’ve defined a topology on a
set . But we also love categories, so we want to see this in
terms of categories. And, indeed, every topology is a category!
First, remember that the collection of subsets of , like the
collection of subobjects on an object in any category, is
partially ordered by inclusion. And since every partially
ordered set is a category, so is the collection of subsets of .
In fact, it’s a lattice, since we can use union and intersection
as our join and meet, respectively. When we say that a poset has
pairwise least upper bounds it’s the same as saying when we
consider it as a category it has finite coproducts, and
similarly pairwise greatest lower bounds are the same as finite
products. But here we can actually take the union or
intersection of any collection of subsets and get a subset, so
we have all products and coproducts. In the language of posets,
we have a “complete lattice”. So now we want to talk about
topologies. A topology is just a collection o