What is wrong with the term "indiscrete topology"? I have noticed recently e.g., in this question that the term "trivial topology" is being used in place of (what I believe to be) the traditional term "indiscrete topology". Is there any reason for this change? To me, the discrete and indiscrete topologies are both trivial, so this change seems like a confusing and retrograde step.
 A: The discrete and indiscrete topologies aren't quite equally trivial from the localic point of view; the indiscrete topology, viewed in terms of its open sets, has the minimum open sets required by the axioms (just $\emptyset$ and the whole set), so arguably it is the "freest" or "most trivial" topology on a fixed set. The corresponding locale (which is the same for all indiscrete spaces; note that discrete spaces by contrast all correspond to different locales) is the terminal locale, which roughly corresponds to the point being the terminal topological space.
A: Here's a small argument in favor of the term "trivial" in this context - or at least, for the implicit claim that the indiscrete topology is more trivial than the discrete topology.
I'd argue that any reasonable notion of "triviality" should be preserved under the standard constructions - namely quotients, substructures (or in this case subspaces), and products. Now while it's true that quotients and substructures of discrete spaces are again discrete, (infinitary) products of discrete spaces need not be discrete and can in fact be quite interesting. By contrast, products of indiscrete spaces remain indiscrete.
Wait, what about coproducts? That's a fair point, but I think - duality notwithstanding - the naive intuition behind triviality plays better with products than coproducts. On a hopefully-ameliorating note, I'd be happy referring to the discrete topology as "cotrivial."
There is probably a more sophisticated observation to be made here along the same lines in terms of the two functors $$\mathscr{Dis}, \mathscr{Ind}:{\bf Sets}\rightarrow{\bf Top}$$ sending a set $X$ to the discrete and indiscrete topology on $X$ respectively; not being a category theorist, however, I'll content myself with the relatively elementary observation above. (EDIT: as Qiaochu Yuan commented, I'm actually making this seem more mysterious than it is: the point is just that these are the left and right adjoints of the forgetful functor ${\bf Top}\rightarrow{\bf Sets}$ respectively and so preserve colimits and limits respectively.)

EDIT: Another justification comes from logic. In logic, we often think of a topology on a set $X$ as corresponding to some notion of "information" about elements of $X$; in particular, the (basic) open sets in the topology correspond to the "atomic observations" that our type of information provides us with.
An indiscrete topology corresponds to no information whatsoever (all points look identical), while a discrete topology corresponds to as much information as possible. Only the former makes sense, to me at least, as a kind of triviality.

In general, as I've gone through point-set topology from various perspectives, I've found more and more reasons to think of indiscrete topologies as trivial and discrete topologies as not-so-trivial. Adopting this terminology from the get-go would, in my opinion, constitute useful terminological foreshadowing.
