Universal definition of the indiscrete topology

olivierd

Jul 17, 2017 5 min read

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 $\catSet^R$ .

exercise