TITLE: Are the discrete and trivial topologies dual in some category theory sense?
QUESTION [6 upvotes]: Wikipedia's page about the trivial topology says it is also commonly called the codiscrete topology. In it, it also says

In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.

This leads to the question: does the codiscrete result from reversing arrows on some category from the discrete topology?

REPLY [8 votes]: Let $\Gamma:\mathbf{Top} \to \mathbf{Set}$ be the forgetful functor. Then, $\Gamma$ has a left adjoint $\text{Disc}$, which equips a set with the discrete topology, and a right adjoint $\text{Codisc}$, which equips a set with the trivial (or codiscrete category).
See nlab's page for more details.