Universal definition of the indiscrete topology
Exercise 182 (Chapter 29, “The category of topological spaces”):
Give a universal definition which leads to the introduction of the indiscrete topology on a set.
On a set , the indiscrete topology is the unique topology such that for any set , any mapping is continuous.
This isn’t really a universal definition. We have seen that the discrete topology can be defined as the unique topology that makes a free topological space on the set . We will give the indiscrete topology, which is at the other end of the spectrum of topologies, a similar definition.
For any category , we can define a “reversed arrows” category , which has the same objects as and the same morphisms, except that the sources and targets of the morphisms are reversed. Composition too is, obviously, reversed; the identities are the same. It is easily checked that this does lead to a proper category, and also that the arrows-reversed category of an arrows-reversed category leads back to the original category.
If we have categories and and a covariant functor from to , then we can consider the three modified functors from to , from to and from to . These functors will map both objects and morphisms exactly as the original does. Those with only one reversal are contravariant; the one with a reversal in both categories is covariant, and is the one that will interest us here.
We consider the classical category of sets ; in its arrows-reversed version, , the morphisms from set to set are the mappings from to . Similarly, we have the category of topological spaces and its arrows-reversed version in which morphisms from topological space to topological space are the continuous mappings from to.
For a given set , object of , we can ask if we can find a free object in category following the forgetful functor from to . This mens finding a topological space and a -morphism such that for any topological space and any -morphism , there exists a unique -morphism such that (as -morphisms) .
Factoring in the arrows-reversals, this means that we wish to find a topological space and a mapping such that for any topological space and any mapping , there exists a unique continuous mapping such that (as mappings) .
We propose that the topological space with the indiscrete topology on , together with the mapping , is a free object on following the functor .
Let be a topological space and a mapping . If is to be a continuous mapping such that , it must be a mapping such that ; that is, we must have . This is indeed continuous, since is the indiscrete topology. Hence it is the unique continuous mapping such that .
Thus we have shown that with the indiscrete topology, together with the identity mapping on , is a free topological space following the arrows-reverse categorical definition of free objects.
Free objects are unique up to an isomorphism. However, we wish to show more specifically that given the set , the indiscrete topology on is the unique topology such that be a free object on following the above definition. We could use the “unique up to an isomorphism” card, but more simply: If is to be a free object on following the above definition, in particular, taking and , we need there to exist a continuous , that is, such that , that is such that . In other words, we need to be continuous from to , which implies , that is, that must be coarser than ; which is possible only if .
Hence the indiscrete topology on is the unique topology such that be a free object in category following the functor from category to category .