Limits of Topological Spaces
We’ve defined topological spaces and continuous maps between them. Together these give us a category . We’d like to understand a few of our favorite categorical constructions as they work in this context.
First off, the empty set has a unique topology, since it only has the one subset at all. Given any other space (we’ll omit explicit mention of its topology) there is a unique function , and it is continuous since the preimage of any subset of is empty. Thus is the initial object in .
On the other side, any singleton set also has a unique topology, since the only subsets are the whole set and the empty set, which must both be open. Given any other space there is a unique function , and it is continuous because the preimage of the empty set is empty and the preimage of the single point is the whole of , both of which are open in . Thus is a terminal object in .
Now for products. Given a family of topological spaces indexed by , we can form the product set , which comes with projection functions and satisfies a universal property. We want to use this same set and these same functions to describe the product in , so we must choose our topology on the product set so that these projections will be continuous. Given an open set , then, its preimage must be open in . Let’s take these preimages to be a subbase and consider the topology they generate.
If is any other space with a family of continuous maps , then the universal property in gives us a unique function . But will it be a continuous map? To check this, remember that we only need to verify it on a subbase for the topology on the product space, and we have one ready to work with. Each set in the subbase is the preimage of an open set in some , and then its preimage under is , which is open by the assumption that each is continuous. And so the product set equipped with the product topology described above is the categorical product of the topological spaces .
What about coproducts? Let’s again start with the coproduct in , which is the disjoint union , and which comes with canonical injections . This time let’s jump right into the universal property, which says that given another space and functions , we have a unique function . Now we need any function we get like this to be continuous. The preimage of an open set will be the union of the preimages of each of the , sitting inside the disjoint union. By choosing , the , and judiciously, we can get the preimage to be any open set we want in , so the open sets in the disjoint union should consist precisely of those subsets whose preimage is open for each . It’s easy to verify that this collection is actually a topology, which then gives us the categorical coproduct in .
If we start with a topological space and take any subset then we can ask for the coarsest topology on that makes the inclusion map continuous, sort of like how we defined the product topology above. The open sets in will be any set of the form for an open subset . Then given another space , a function will be continuous if and only if is continuous. Indeed, the preimage clearly shows this equivalence. We call this the subspace topology on .
In particular, if we have two continuous maps and , then we can consider the subspace consisting of those points satisfying . Given any other space and a continuous map such that , clearly sends all of into the set ; the function factors as , where is the inclusion map. Then must be continuous because is, and so the subspace is the equalizer of the maps and .
Dually, given a topological space and an equivalence relation on the underlying set of we can define the quotient space to be the set of equivalence classes of points of . This comes with a canonical function , which we want to be continuous. Further, we know that if is any function for which implies , then factors as for some function . We want to define the topology on the quotient set so that is continuous if and only if is. Given an open set , its preimage is the set of equivalence classes that get sent into , while its preimage is the set of all points that get sent to . And so we say a subset of the quotient space is open if and only if its preimage — the union of the equivalence classes in is open in .
In particular, if we have two maps and we get an equivalence relation on by defining if there is a so that and . If we walk through the above description of the quotient space we find that this construction gives us the coequalizer of and .
And now, the existence theorem for limits tells us that all limits and colimits exist in . That is, the category of topological spaces is both complete and cocomplete.
As a particularly useful example, let’s look at an example of a pushout. If we have two topological spaces and and a third space with maps and making into a subspace of both and , then we can construct the pushout of and over . The general rule is to first construct the coproduct of and , and then pass to an appropriate coequalizer. That is, we take the disjoint union and then identify the points in the copy of sitting inside with those in the copy of sitting inside . That is, we get the union of and , “glued” along .