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 X, the indiscrete topology is the unique topology such that for any set Y, any mapping \phi: Y \to X is continuous.

This isn’t really a universal definition. We have seen that the discrete topology \tau_D can be defined as the unique topology \tau that makes ((X, \tau), \Id_X) a free topological space on the set X. We will give the indiscrete topology, which is at the other end of the spectrum of topologies, a similar definition.

For any category \mathfrak C, we can define a “reversed arrows” category \mathfrak C^R, which has the same objects as \mathfrak C 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 \mathfrak C and \mathfrak D and a covariant functor \mathcal F from \mathfrak D to \mathfrak C, then we can consider the three modified functors \mathcal F^{DR} from \mathfrak D to \mathfrak C^R, \mathcal F^{RD} from \mathfrak D^R to \mathfrak C and\mathcal F^{RR} from \mathfrak D^R to \mathfrak C^R. These functors will map both objects and morphisms exactly as the original \mathcal F 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 \catSet; in its arrows-reversed version, \catSet^R, the morphisms from set A to set B are the mappings from B to A. Similarly, we have the category of topological spaces \catTop and its arrows-reversed version \catTop^R in which morphisms from topological space (X, \tau) to topological space (Y, \sigma) are the continuous mappings from(Y, \sigma) to(X, \tau).

For a given set A, object of \catSet^R, we can ask if we can find a free object (X, \tau) in category \catTop^R following the forgetful functor \mathcal F^{RR} from \catTop^R to \catSet^R. This mens finding a topological space (X, \tau) and a \catSet^R-morphism \alpha: A \to \mathcal F^{RR}((X, \tau)) = X such that for any topological space (Y, \sigma) and any \catSet^R-morphism \alpha': A \to \mathcal F^{RR}((Y, \sigma)) = Y, there exists a unique \catTop^R-morphism \gamma: (X, \tau) \to (Y, \sigma) such that (as \catSet^R-morphisms) \alpha' = \gamma \circ \alpha.

Factoring in the arrows-reversals, this means that we wish to find a topological space (X, \tau) and a mapping \alpha: X \to A such that for any topological space (Y, \sigma) and any mapping \alpha': Y \to A, there exists a unique continuous mapping \gamma: (Y, \sigma) \to (X, \tau) such that (as mappings) \alpha' = \alpha \circ \gamma.

We propose that the topological space (A, \tau_I) with \tau_I the indiscrete topology on A, together with the mapping \alpha = \Id_A, is a free object on A following the functor \mathcal F^{RR}.

Proof:

Let (Y, \sigma) be a topological space and \alpha' a mapping Y \to A. If \gamma is to be a continuous mapping (Y, \sigma) \to (A, \tau_I) such that \alpha' = \alpha \circ \gamma, it must be a mapping Y \to A such that \alpha' = \Id_A \circ \gamma = \gamma; that is, we must have \gamma = \alpha'. This \gamma is indeed continuous, since \tau_I is the indiscrete topology. Hence it is the unique continuous mapping (Y, \sigma) \to (A, \tau_I) such that \alpha' = \alpha \circ \gamma.

Thus we have shown that A with the indiscrete topology, together with the identity mapping on A, 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 A, the indiscrete topology \tau_I on A is the unique topology \tau such that ((A, \tau), \Id_A) be a free object on A following the above definition. We could use the “unique up to an isomorphism” card, but more simply: If ((A, \tau), \Id_A) is to be a free object on A following the above definition, in particular, taking (Y, \sigma) = (A, \tau_I) and \alpha' = \Id_A, we need there to exist a continuous \gamma: (Y, \sigma) \to (A, \tau), that is, (A, \tau_I) \to (A, \tau) such that \alpha' = \Id_A \circ \gamma, that is such that \gamma = \Id_A. In other words, we need \Id_A to be continuous from (A, \tau_I) to (A, \tau), which implies \tau \subseteq \tau_I, that is, that \tau must be coarser than \tau_I; which is possible only if \tau = \tau_I.

Hence the indiscrete topology on A is the unique topology \tau such that ((A, \tau), \Id_A) be a free object in category \catTop^R following the functor \mathcal F^{RR} from category \catTop^R to category \catSet^R.

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